Content check — Intermediate Algebra, Chapter 4: Systems of Linear Equations

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 4: Systems of Linear Equations

351 problems · 60 figures

Be Prepared 4.1 §4.1 source ↗

For the equation

y = \frac{2}{3} x - 4,

ⓐ Is

(6, 0)

a solution? ⓑ Is

(−3, −2)

a solution?
If you missed this problem, review Example 3.2.

Example 3.2 in the book ↗

Answer

ⓐ yes; ⓑ no

Be Prepared 4.2 §4.1 source ↗

Find the slope and y-intercept of the line

3 x - y = 12 .

If you missed this problem, review Example 3.16.

Example 3.16 in the book ↗

Answer

m = 3; b = - 12

Be Prepared 4.3 §4.1 source ↗

Find the x- and y-intercepts of the line

2 x - 3 y = 12 .

If you missed this problem, review Example 3.8.

Example 3.8 in the book ↗

Answer

6, 0, 0, - 4

Try It 4.1 §4.1 source ↗

Determine whether the ordered pair is a solution to the system $\begin{matrix} {\begin{matrix} 3 x + y = 0 \\ x + 2 y = −5 \end{matrix} . \end{matrix}$

ⓐ $(1, −3)$ ⓑ $(0, 0)$

Answer

ⓐ yes ⓑ no

Try It 4.2 §4.1 source ↗

Determine whether the ordered pair is a solution to the system $\begin{matrix} {\begin{matrix} x - 3 y = −8 \\ - 3 x - y = 4 \end{matrix} . \end{matrix}$

ⓐ $(2, −2)$ ⓑ $(−2, 2)$

Answer

ⓐ no ⓑ yes

Try It 4.3 §4.1 source ↗

Solve the system by graphing:

{\begin{matrix} x - 3 y = −3 \\ x + y = 5 \end{matrix} .

Answer

(3, 2)

Try It 4.4 §4.1 source ↗

Solve the system by graphing:

{\begin{matrix} - x + y = 1 \\ 3 x + 2 y = 12 \end{matrix} .

Answer

(2, 3)

Try It 4.5 §4.1 source ↗

Solve the system by graphing:

{\begin{matrix} - x + y = 1 \\ 2 x + y = 10 \end{matrix} .

Answer

(3, 4)

Try It 4.6 §4.1 source ↗

Solve the system by graphing:

{\begin{matrix} 2 x + y = 6 \\ x + y = 1 \end{matrix} .

Answer

(5, −4)

Try It 4.7 §4.1 source ↗

Solve the system by graphing:

{\begin{matrix} y = - \frac{1}{4} x + 2 \\ x + 4 y = - 8 \end{matrix} .

Answer

no solution

Try It 4.8 §4.1 source ↗

Solve the system by graphing:

{\begin{matrix} y = 3 x - 1 \\ 6 x - 2 y = 6 \end{matrix} .

Answer

no solution

Try It 4.9 §4.1 source ↗

Solve the system by graphing:

{\begin{matrix} y = - 3 x - 6 \\ 6 x + 2 y = - 12 \end{matrix} .

Answer

infinitely many solutions

Try It 4.10 §4.1 source ↗

Solve the system by graphing:

{\begin{matrix} y = \frac{1}{2} x - 4 \\ 2 x - 4 y = 16 \end{matrix} .

Answer

infinitely many solutions

Try It 4.11 §4.1 source ↗

Without graphing, determine the number of solutions and then classify the system of equations.

ⓐ ${\begin{matrix} y = −2 x - 4 \\ 4 x + 2 y = 9 \end{matrix}$ ⓑ ${\begin{matrix} 3 x + 2 y = 2 \\ 2 x + y = 1 \end{matrix}$

Answer

ⓐ no solution, inconsistent, independent ⓑ one solution, consistent, independent

Try It 4.12 §4.1 source ↗

Without graphing, determine the number of solutions and then classify the system of equations.

ⓐ ${\begin{matrix} y = \frac{1}{3} x - 5 \\ x - 3 y = 6 \end{matrix}$ ⓑ ${\begin{matrix} x + 4 y = 12 \\ - x + y = 3 \end{matrix}$

Answer

ⓐ no solution, inconsistent, independent ⓑ one solution, consistent, independent

Try It 4.13 §4.1 source ↗

Solve the system by substitution:

{\begin{matrix} - 2 x + y = −11 \\ x + 3 y = 9 \end{matrix} .

Answer

(6, 1)

Try It 4.14 §4.1 source ↗

Solve the system by substitution:

{\begin{matrix} 2 x + y = −1 \\ 4 x + 3 y = 3 \end{matrix} .

Answer

(−3, 5)

Try It 4.15 §4.1 source ↗

Solve the system by substitution:

{\begin{matrix} x - 4 y = −4 \\ - 3 x + 4 y = 0 \end{matrix} .

Answer

(2, \frac{3}{2})

Try It 4.16 §4.1 source ↗

Solve the system by substitution:

{\begin{matrix} 4 x - y = 0 \\ 2 x - 3 y = 5 \end{matrix} .

Answer

(- \frac{1}{2}, −2)

Try It 4.17 §4.1 source ↗

Solve the system by elimination:

{\begin{matrix} 3 x + y = 5 \\ 2 x - 3 y = 7 \end{matrix} .

Answer

(2, −1)

Try It 4.18 §4.1 source ↗

Solve the system by elimination:

{\begin{matrix} 4 x + y = - 5 \\ - 2 x - 2 y = - 2 \end{matrix} .

Answer

(−2, 3)

Try It 4.19 §4.1 source ↗

Solve the system by elimination:

{\begin{matrix} 3 x - 4 y = - 9 \\ 5 x + 3 y = 14 \end{matrix} .

Answer

(1, 3)

Try It 4.20 §4.1 source ↗

Solve each system by elimination:

{\begin{matrix} 7 x + 8 y = 4 \\ 3 x - 5 y = 27 \end{matrix} .

Answer

(4, −3)

Try It 4.21 §4.1 source ↗

Solve each system by elimination:

{\begin{matrix} \frac{1}{3} x - \frac{1}{2} y = 1 \\ \frac{3}{4} x - y = \frac{5}{2} \end{matrix} .

Answer

(6, 2)

Try It 4.22 §4.1 source ↗

Solve each system by elimination:

{\begin{matrix} x + \frac{3}{5} y = - \frac{1}{5} \\ - \frac{1}{2} x - \frac{2}{3} y = \frac{5}{6} \end{matrix} .

Answer

(1, −2)

Try It 4.23 §4.1 source ↗

Solve the system by elimination:

{\begin{matrix} 5 x - 3 y = 15 \\ y = - 5 + \frac{5}{3} x \end{matrix} .

Answer

infinitely many solutions

Try It 4.24 §4.1 source ↗

Solve the system by elimination:

{\begin{matrix} x + 2 y = 6 \\ y = - \frac{1}{2} x + 3 \end{matrix} .

Answer

infinitely many solutions

Try It 4.25 §4.1 source ↗

For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.

ⓐ ${\begin{matrix} 4 x - 5 y = −32 \\ 3 x + 2 y = −1 \end{matrix}$ ⓑ ${\begin{matrix} x = 2 y - 1 \\ 3 x - 5 y = −7 \end{matrix}$

Answer

ⓐ Since both equations are in standard form, using elimination will be most convenient. ⓑ Since one equation is already solved for x, using substitution will be most convenient.

Try It 4.26 §4.1 source ↗

For each system of linear equations decide whether it would be more convenient to solve it by substitution or elimination. Explain your answer.

ⓐ ${\begin{matrix} y = 2 x - 1 \\ 3 x - 4 y = - 6 \end{matrix}$ ⓑ ${\begin{matrix} 6 x - 2 y = 12 \\ 3 x + 7 y = −13 \end{matrix}$

Answer

ⓐ Since one equation is already solved for y, using substitution will be most convenient. ⓑ Since both equations are in standard form, using elimination will be most convenient.

4.1-1 §4.1 source ↗

Determine Whether an Ordered Pair is a Solution of a System of Equations

In the following exercises, determine if the following points are solutions to the given system of equations.

${\begin{matrix} 2 x - 6 y = 0 \\ 3 x - 4 y = 5 \end{matrix}$

ⓐ $(3, 1)$ ⓑ $(−3, 4)$

Answer

ⓐ yes ⓑ no

4.1-3 §4.1 source ↗

Determine Whether an Ordered Pair is a Solution of a System of Equations

In the following exercises, determine if the following points are solutions to the given system of equations.

${\begin{matrix} x + y = 2 \\ y = \frac{3}{4} x \end{matrix}$

ⓐ $(\frac{8}{7}, \frac{6}{7})$ ⓑ $(1, \frac{3}{4})$

Answer

ⓐ yes ⓑ no

4.1-5 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} 3 x + y = −3 \\ 2 x + 3 y = 5 \end{matrix}$

Answer

(−2, 3)

4.1-7 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} y = x + 2 \\ y = −2 x + 2 \end{matrix}$

Answer

(0, 2)

4.1-9 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} y = \frac{3}{2} x + 1 \\ y = - \frac{1}{2} x + 5 \end{matrix}$

Answer

(2, 4)

4.1-11 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} x + y = −4 \\ - x + 2 y = −2 \end{matrix}$

Answer

(−2, −2)

4.1-13 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} - 2 x + 3 y = 3 \\ x + 3 y = 12 \end{matrix}$

Answer

(3, 3)

4.1-15 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} x + 3 y = −6 \\ y = - \frac{4}{3} x + 4 \end{matrix}$

Answer

(6, −4)

4.1-17 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} - 2 x + 4 y = 4 \\ y = \frac{1}{2} x \end{matrix}$

Answer

no solution

4.1-19 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} 4 x - 3 y = 8 \\ 8 x - 6 y = 14 \end{matrix}$

Answer

no solution

4.1-21 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} x = −3 y + 4 \\ 2 x + 6 y = 8 \end{matrix}$

Answer

infinite solutions

4.1-23 §4.1 source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} 2 x + y = 6 \\ - 8 x - 4 y = −24 \end{matrix}$

Answer

infinite solutions

4.1-25 §4.1 source ↗

Without graphing, determine the number of solutions and then classify the system of equations.

${\begin{matrix} y = \frac{2}{3} x + 1 \\ - 2 x + 3 y = 5 \end{matrix}$

Answer

No solution, Inconsistent, Independent

4.1-27 §4.1 source ↗

Without graphing, determine the number of solutions and then classify the system of equations.

${\begin{matrix} 5 x + 3 y = 4 \\ 2 x - 3 y = 5 \end{matrix}$

Answer

1 point, consistent and independent

4.1-29 §4.1 source ↗

Without graphing, determine the number of solutions and then classify the system of equations.

${\begin{matrix} 5 x - 2 y = 10 \\ y = \frac{5}{2} x - 5 \end{matrix}$

Answer

infinite solutions, consistent, dependent

4.1-31 §4.1 source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} 2 x + y = −2 \\ 3 x - y = 7 \end{matrix}$

Answer

(1, −4)

4.1-33 §4.1 source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} x - 3 y = −9 \\ 2 x + 5 y = 4 \end{matrix}$

Answer

(−3, 2)

4.1-35 §4.1 source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} - 2 x + 2 y = 6 \\ y = −3 x + 1 \end{matrix}$

Answer

(- \frac{1}{2}, \frac{5}{2})

4.1-37 §4.1 source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} 3 x + 4 y = 1 \\ y = - \frac{2}{5} x + 2 \end{matrix}$

Answer

(−5, 4)

4.1-39 §4.1 source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} 4 x + y = 10 \\ x - 2 y = −20 \end{matrix}$

Answer

(0, 10)

4.1-41 §4.1 source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} y = x - 6 \\ y = - \frac{3}{2} x + 4 \end{matrix}$

Answer

(4, −2)

4.1-43 §4.1 source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} 2 x - 16 y = 8 \\ - x - 8 y = −4 \end{matrix}$

Answer

(4, 0)

4.1-45 §4.1 source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} y = - \frac{2}{3} x + 5 \\ 2 x + 3 y = 11 \end{matrix}$

Answer

no solution

4.1-47 §4.1 source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

${\begin{matrix} 6 x - 5 y = −1 \\ 2 x + y = 13 \end{matrix}$

Answer

(4, 5)

4.1-49 §4.1 source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

${\begin{matrix} 5 x - 3 y = −1 \\ 2 x - y = 2 \end{matrix}$

Answer

(7, 12)

4.1-51 §4.1 source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

${\begin{matrix} 4 x - 3 y = 3 \\ 2 x + 5 y = −31 \end{matrix}$

Answer

(−3, −5)

4.1-53 §4.1 source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

${\begin{matrix} 11 x + 9 y = −5 \\ 7 x + 5 y = −1 \end{matrix}$

Answer

(2, −3)

4.1-55 §4.1 source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

${\begin{matrix} 2 x + 9 y = −4 \\ 3 x + 13 y = −7 \end{matrix}$

Answer

(−11, 2)

4.1-57 §4.1 source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

${\begin{matrix} x + \frac{1}{2} y = \frac{3}{2} \\ \frac{1}{5} x - \frac{1}{5} y = 3 \end{matrix}$

Answer

(6, - 9)

4.1-59 §4.1 source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

${\begin{matrix} \frac{1}{3} x - y = −3 \\ \frac{2}{3} x + \frac{5}{2} y = 3 \end{matrix}$

Answer

(−3, 2)

4.1-61 §4.1 source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

${\begin{matrix} x - 4 y = −1 \\ - 3 x + 12 y = 3 \end{matrix}$

Answer

infinitely many

4.1-63 §4.1 source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

${\begin{matrix} 4 x + 3 y = 2 \\ 20 x + 15 y = 10 \end{matrix}$

Answer

infinitely many

4.1-65 §4.1 source ↗

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

ⓐ ${\begin{matrix} y = 7 x - 5 \\ 3 x - 2 y = 16 \end{matrix}$
ⓑ ${\begin{matrix} 12 x - 5 y = −42 \\ 3 x + 7 y = −15 \end{matrix}$

Answer

ⓐ substitution ⓑ elimination

4.1-67 §4.1 source ↗

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

ⓐ ${\begin{matrix} 14 x - 15 y = −30 \\ 7 x + 2 y = 10 \end{matrix}$
ⓑ ${\begin{matrix} x = 9 y - 11 \\ 2 x - 7 y = −27 \end{matrix}$

Answer

ⓐ elimination ⓑ substituion

4.1-69 §4.1 source ↗

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

Solve the system of equations by substitution and explain all your steps in words: ${\begin{matrix} 3 x + y = 12 \\ x = y - 8 \end{matrix} .$

Answer

Answers will vary.

4.1-71 §4.1 source ↗

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

Solve the system of equations ${\begin{matrix} x + y = 10 \\ x - y = 6 \end{matrix}$

ⓐ by graphing ⓑ by substitution
ⓒ Which method do you prefer? Why?

Answer

Answers will vary.

Be Prepared 4.4 §4.2 source ↗

The sum of twice a number and nine is 31. Find the number.
If you missed this problem, review Example 2.15.

Example 2.15 in the book ↗

Answer

11

Be Prepared 4.5 §4.2 source ↗

Twins Jon and Ron together earned $96,000 last year. Ron earned $8000 more than three times what Jon earned. How much did each of the twins earn?
If you missed this problem, review Example 2.19.

Example 2.19 in the book ↗

Answer

Jon earned $22,000 and Ron earned $74,000.

Be Prepared 4.6 §4.2 source ↗

An express train and a local train leave Pittsburgh to travel to Washington, D.C. The express train can make the trip in four hours and the local train takes five hours for the trip. The speed of the express train is 12 miles per hour faster than the speed of the local train. Find the speed of both trains.
If you missed this problem, review Example 2.43.

Example 2.43 in the book ↗

Answer

The speed of the local train is 48 mph and the speed of the express train is 60 mph.

Try It 4.27 §4.2 source ↗

The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

Answer

3, 7

Try It 4.28 §4.2 source ↗

The sum of two numbers is

−6 .

One number is 10 less than the other. Find the numbers.

Answer

−8

Try It 4.29 §4.2 source ↗

Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. The second pays a salary of $20,000 plus a commission of $50 for each policy sold. How many policies would need to be sold to make the total pay the same?

Answer

160 policies

Try It 4.30 §4.2 source ↗

Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?

Answer

1000 suits

Try It 4.31 §4.2 source ↗

Translate to a system of equations and then solve:

Mark went to the gym and did 40 minutes of Bikram hot yoga and 10 minutes of jumping jacks. He burned 510 calories. The next time he went to the gym, he did 30 minutes of Bikram hot yoga and 20 minutes of jumping jacks burning 470 calories. How many calories were burned for each minute of yoga? How many calories were burned for each minute of jumping jacks?

Answer

Mark burned 11 calories for each minute of yoga and 7 calories for each minute of jumping jacks.

Try It 4.32 §4.2 source ↗

Translate to a system of equations and then solve:

Erin spent 30 minutes on the rowing machine and 20 minutes lifting weights at the gym and burned 430 calories. During her next visit to the gym she spent 50 minutes on the rowing machine and 10 minutes lifting weights and burned 600 calories. How many calories did she burn for each minutes on the rowing machine? How many calories did she burn for each minute of weight lifting?

Answer

Erin burned 11 calories for each minute on the rowing machine and 5 calories for each minute of weight lifting.

Try It 4.33 §4.2 source ↗

Translate to a system of equations and then solve:

The difference of two complementary angles is 20 degrees. Find the measures of the angles.

Answer

The angle measures are 55 and 35.

Try It 4.34 §4.2 source ↗

Translate to a system of equations and then solve:

The difference of two complementary angles is 80 degrees. Find the measures of the angles.

Answer

The angle measures are 5 and 85.

Try It 4.35 §4.2 source ↗

Translate to a system of equations and then solve:

Two angles are supplementary. The measure of the larger angle is 12 degrees more than three times the smaller angle. Find the measures of the angles.

Answer

The angle measures are 42 and 138.

Try It 4.36 §4.2 source ↗

Translate to a system of equations and then solve:

Two angles are supplementary. The measure of the larger angle is 18 less than twice the measure of the smaller angle. Find the measures of the angles.

Answer

The angle measures are 66 and 114.

Try It 4.37 §4.2 source ↗

The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

Answer

22, 68

Try It 4.38 §4.2 source ↗

The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

Answer

36, 54

Try It 4.39 §4.2 source ↗

Translate to a system of equations and then solve:

Mario wants to put a fence around the pool in his backyard. Since one side is adjacent to the house, he will only need to fence three sides. There are two long sides and the one shorter side is parallel to the house. He needs 155 feet of fencing to enclose the pool. The length of the long side is 10 feet less than twice the width. Find the length and width of the pool area to be enclosed.

Answer

The length is 60 feet and the width is 35 feet.

Try It 4.40 §4.2 source ↗

Translate to a system of equations and then solve:

Alexis wants to build a rectangular dog run in her yard adjacent to her neighbor’s fence. She will use 136 feet of fencing to completely enclose the rectangular dog run. The length of the dog run along the neighbor’s fence will be 16 feet less than twice the width. Find the length and width of the dog run.

Answer

The length is 60 feet and the width is 38 feet.

Try It 4.41 §4.2 source ↗

Translate to a system of equations and then solve:

Mitchell left Detroit on the interstate driving south towards Orlando at a speed of 60 miles per hour. Clark left Detroit 1 hour later traveling at a speed of 75 miles per hour, following the same route as Mitchell. How long will it take Clark to catch Mitchell?

Answer

It will take Clark 4 hours to catch Mitchell.

Try It 4.42 §4.2 source ↗

Translate to a system of equations and then solve:

Charlie left his mother’s house traveling at an average speed of 36 miles per hour. His sister Sally left 15 minutes $(\frac{1}{4} hour)$ later traveling the same route at an average speed of 42 miles per hour. How long before Sally catches up to Charlie?

Answer

It will take Sally

1 \frac{1}{2}

hours to catch up to Charlie.

Try It 4.43 §4.2 source ↗

Translate to a system of equations and then solve:

A Mississippi river boat cruise sailed 120 miles upstream for 12 hours and then took 10 hours to return to the dock. Find the speed of the river boat in still water and the speed of the river current.

Answer

The rate of the boat is 11 mph and the rate of the current is 1 mph.

Try It 4.44 §4.2 source ↗

Translate to a system of equations and then solve:

Jason paddled his canoe 24 miles upstream for 4 hours. It took him 3 hours to paddle back. Find the speed of the canoe in still water and the speed of the river current.

Answer

The speed of the canoe is 7 mph and the speed of the current is 1 mph.

Try It 4.45 §4.2 source ↗

Translate to a system of equations and then solve:

A small jet can fly 1,325 miles in 5 hours with a tailwind but only 1,035 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Answer

The speed of the jet is 236 mph and the speed of the wind is 29 mph.

Try It 4.46 §4.2 source ↗

Translate to a system of equations and then solve:

A commercial jet can fly 1,728 miles in 4 hours with a tailwind but only 1,536 miles in 4 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Answer

The speed of the jet is 408 mph and the speed of the wind is 24 mph.

4.2-73 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

The sum of two number is 30. One number is 4 less than the other. Find the numbers.

Answer

13 and 17

4.2-75 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

The sum of two number is $−26 .$ One number is 12 less than the other. Find the numbers.

Answer

−7

and

−19

4.2-77 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

The sum of two numbers is 37. Their difference is 9. Find the numbers.

Answer

14 and 23

4.2-79 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

The sum of two numbers is $−45 .$ Their difference is $−89 .$ Find the numbers.

Answer

22 and

−67

4.2-81 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Jackie has been offered positions by two cable companies. The first company pays a salary of $14,000 plus a commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?

Answer

Eighty cable packages would need to be sold to make the total pay the same.

4.2-83 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?

Answer

Mitchell would need to sell 120 stoves for the companies to be equal.

4.2-85 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?

Answer

8 and 40 gallons

4.2-87 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Drew burned 1800 calories Friday playing one hour of basketball and canoeing for two hours. Saturday he spent two hours playing basketball and three hours canoeing and burned 3200 calories. How many calories did he burn per hour when playing basketball? How many calories did he burn per hour when canoeing?

Answer

1000 calories playing basketball and 400 calories canoeing

4.2-89 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Nancy bought seven pounds of oranges and three pounds of bananas for $17. Her husband later bought three pounds of oranges and six pounds of bananas for $12. What was the cost per pound of the oranges and the bananas?

Answer

Oranges cost $2 per pound and bananas cost $1 per pound

4.2-91 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Peter is buying office supplies. He is able to buy 3 packages of paper and 4 staplers for $40 or he is able to buy 5 packages of paper and 6 staplers for $62. How much does a package of paper cost? How much does a stapler cost?

Answer

Package of paper $4, stapler $7

4.2-93 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

The total number of calories in 2 hot dogs and 3 cups of cottage cheese is 960 calories. The total number of calories in 5 hot dogs and 2 cups of cottage cheese is 1190 calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?

Answer

Hot dog 150 calories, cup of cottage cheese 220 calories

4.2-95 §4.2 source ↗

Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Owen is making lemonade from concentrate. The number of quarts of water he needs is 4 times the number of quarts of concentrate. How many quarts of water and how many quarts of concentrate does Owen need to make 100 quarts of lemonade?

Answer

Owen will need 80 quarts of water and 20 quarts of concentrate to make 100 quarts of lemonade.

4.2-97 §4.2 source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

The difference of two complementary angles is 17 degrees. Find the measures of the angles.

Answer

53.5

degrees and

36.5

degrees

4.2-99 §4.2 source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

Two angles are complementary. The measure of the larger angle is ten more than four times the measure of the smaller angle. Find the measures of both angles.

Answer

16 degrees and 74 degrees

4.2-101 §4.2 source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

The difference of two supplementary angles is 88 degrees. Find the measures of the angles.

Answer

134 degrees and 46 degrees

4.2-103 §4.2 source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

Two angles are supplementary. The measure of the larger angle is five less than four times the measure of the smaller angle. Find the measures of both angles.

Answer

37 degrees and 143 degrees

4.2-105 §4.2 source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.

Answer

16 degrees and 74 degrees

4.2-107 §4.2 source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.

Answer

45 degrees and 45 degrees

4.2-109 §4.2 source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his front yard. The length is five feet less than three times the width. Find the length and width of the fencing.

Answer

Width is 41 feet and length is 118 feet.

4.2-111 §4.2 source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

The perimeter of a toddler play area is 100 feet. The length is ten more than three times the width. Find the length and width of the play area.

Answer

Width is 10 feet and length is 40 feet.

4.2-113 §4.2 source ↗

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

College roommates John and David were driving home to the same town for the holidays. John drove 55 mph, and David, who left an hour later, drove 60 mph. How long will it take for David to catch up to John?

Answer

12 hours

4.2-115 §4.2 source ↗

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

Felecia left her home to visit her daughter driving 45 mph. Her husband waited for the dog sitter to arrive and left home twenty minutes (1/3 hour) later. He drove 55 mph to catch up to Felecia. How long before he reaches her?

Answer

1.83 hour

4.2-117 §4.2 source ↗

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

A motor boat travels 60 miles down a river in three hours but takes five hours to return upstream. Find the rate of the boat in still water and the rate of the current.

Answer

Boat rate is 16 mph and current rate is 4 mph.

4.2-119 §4.2 source ↗

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

A river cruise boat sailed 80 miles down the Mississippi River for four hours. It took five hours to return. Find the rate of the cruise boat in still water and the rate of the current.

Answer

Boat rate is 18 mph and current rate is 2 mph.

4.2-121 §4.2 source ↗

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

A small jet can fly 1435 miles in 5 hours with a tailwind but only 1,215 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Answer

Jet rate is 265 mph and wind speed is 22 mph.

4.2-123 §4.2 source ↗

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

A commercial jet can fly 1,320 miles in 3 hours with a tailwind but only 1170 miles in 3 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Answer

Jet rate is 415 mph and wind speed is 25 mph.

4.2-125 §4.2 source ↗

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

Write a uniform motion problem similar to Example 4.15 that relates to where you live with your friends or family members. Then translate to a system of equations and solve it.

Answer

Answers will vary.

Be Prepared 4.7 §4.3 source ↗

Multiply:

4.025 (1,562) .

If you missed this problem, review Example 1.36.

Example 1.36 in the book ↗

Answer

6,287.05

Be Prepared 4.8 §4.3 source ↗

Write 8.2% as a decimal.
If you missed this problem, review Example 1.40.

Example 1.40 in the book ↗

Answer

0.082

Be Prepared 4.9 §4.3 source ↗

Earl’s dinner bill came to $32.50 and he wanted to leave an 18% tip. How much should the tip be?
If you missed this problem, review Example 2.20.

Example 2.20 in the book ↗

Answer

$5.85

Try It 4.47 §4.3 source ↗

Translate to a system of equations and solve:

The ticket office at the zoo sold 553 tickets one day. The receipts totaled $3,936. How many $9 adult tickets and how many $6 child tickets were sold?

Answer

206 adults, 347 children

Try It 4.48 §4.3 source ↗

Translate to a system of equations and solve:

The box office at a movie theater sold 147 tickets for the evening show, and receipts totaled $1,302. How many $11 adult and how many $8 child tickets were sold?

Answer

42 adults, 105 children

Try It 4.49 §4.3 source ↗

Translate to a system of equations and solve:

Matilda has a handful of quarters and dimes, with a total value of $8.55. The number of quarters is 3 more than twice the number of dimes. How many dimes and how many quarters does she have?

Answer

13 dimes and 29 quarters

Try It 4.50 §4.3 source ↗

Translate to a system of equations and solve:

Priam has a collection of nickels and quarters, with a total value of $7.30. The number of nickels is six less than three times the number of quarters. How many nickels and how many quarters does he have?

Answer

19 quarters and 51 nickels

Try It 4.51 §4.3 source ↗

Translate to a system of equations and solve:

Greta wants to make 5 pounds of a nut mix using peanuts and cashews. Her budget requires the mixture to cost her $6 per pound. Peanuts are $4 per pound and cashews are $9 per pound. How many pounds of peanuts and how many pounds of cashews should she use?

Answer

3 pounds peanuts and 2 pounds cashews

Try It 4.52 §4.3 source ↗

Translate to a system of equations and solve:

Sammy has most of the ingredients he needs to make a large batch of chili. The only items he lacks are beans and ground beef. He needs a total of 20 pounds combined of beans and ground beef and has a budget of $3 per pound. The price of beans is $1 per pound and the price of ground beef is $5 per pound. How many pounds of beans and how many pounds of ground beef should he purchase?

Answer

10 pounds of beans, 10 pounds of ground beef

Try It 4.53 §4.3 source ↗

Translate to a system of equations and solve:

LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?

Answer

120 ml of 25% solution and 30 ml of 50% solution

Try It 4.54 §4.3 source ↗

Translate to a system of equations and solve:

Anatole needs to make 250 milliliters of a 25% solution of hydrochloric acid for a lab experiment. The lab only has a 10% solution and a 40% solution in the storeroom. How much of the 10% and how much of the 40% solutions should he mix to make the 25% solution?

Answer

125 ml of 10% solution and 125 ml of 40% solution

Try It 4.55 §4.3 source ↗

Translate to a system of equations and solve:

Leon had $50,000 to invest and hopes to earn $6.2 %$ interest per year. He will put some of the money into a stock fund that earns 7% per year and the rest in to a savings account that earns 2% per year. How much money should he put into each fund?

Answer

$42,000 in the stock fund and $8000 in the savings account

Try It 4.56 §4.3 source ↗

Translate to a system of equations and solve:

Julius invested $7000 into two stock investments. One stock paid 11% interest and the other stock paid 13% interest. He earned $12.5 %$ interest on the total investment. How much money did he put in each stock?

Answer

$1750 at 11% and $5250 at 13%

Try It 4.57 §4.3 source ↗

Translate to a system of equations and solve:

Laura owes $18,000 on her student loans. The interest rate on the bank loan is 2.5% and the interest rate on the federal loan is 6.9%. The total amount of interest she paid last year was $1,066. What was the principal for each loan?

Answer

Bank $4,000; Federal $14,000

Try It 4.58 §4.3 source ↗

Translate to a system of equations and solve:

Jill’s Sandwich Shoppe owes $65,200 on two business loans, one at 4.5% interest and the other at 7.2% interest. The total amount of interest owed last year was $3,582. What was the principal for each loan?

Answer

$41,200 at 4.5%, $24,000 at 7.2%

Try It 4.59 §4.3 source ↗

The manufacturer of a weight training bench spends $15 to build each bench and sells them for $32. The manufacturer also has fixed costs each month of $25,500.

ⓐ Find the cost function C when x benches are manufactured.

ⓑ Find the revenue function R when x benches are sold.

ⓓ Find the break-even point. Interpret what the break-even point means.

Answer

ⓐ $C (x) =$ $15 x +$ $25,$ $500$

ⓑ $R (x) = 32 x$

ⓒ

Figure shows a graph with two intersecting lines. One of them passes through the origin. The other crosses the y axis at point 25,687.

ⓓ $1,500$ ; when 1,500 benches are sold, the cost and revenue will be both 48,000

Try It 4.60 §4.3 source ↗

The manufacturer of a weight training bench spends $120 to build each bench and sells them for $170. The manufacturer also has fixed costs each month of $150,000.

ⓐ Find the cost function C when x benches are manufactured.

ⓑ Find the revenue function R when x benches are sold.

ⓓ Find the break-even point. Interpret what the break-even point means.

Answer

ⓐ $C (x) =$ $120 x +$ $150,000$

ⓑ $R (x) = 170 x$

ⓒ

Figure shows a graph with two intersecting lines. One of them passes through the origin.

ⓓ $3,000$ ; when 3,000 benches are sold, the revenue and costs are both $510,000

4.3-127 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

Tickets for the Cirque du Soleil show are $70 for adults and $50 for children. One evening performance had a total of 300 tickets sold and the receipts totaled $17,200. How many adult and how many child tickets were sold?

Answer

110 adult tickets, 190 child tickets

4.3-129 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

Tickets for a Minnesota Twins baseball game are $69 for Main Level seats and $39 for Terrace Level seats. A group of sixteen friends went to the game and spent a total of $804 for the tickets. How many of Main Level and how many Terrace Level tickets did they buy?

Answer

6 good seats, 10 cheap seats

4.3-131 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

Tickets for the community fair cost $12 for adults and $5 dollars for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?

Answer

92 adult tickets, 220 children tickets

4.3-133 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

Sherri saves nickels and dimes in a coin purse for her daughter. The total value of the coins in the purse is $$ 0.95 .$ The number of nickels is two less than five times the number of dimes. How many nickels and how many dimes are in the coin purse?

Answer

13 nickels, 3 dimes

4.3-135 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

Lucinda had a pocketful of dimes and quarters with a value of $$ 6.20 .$ The number of dimes is eighteen more than three times the number of quarters. How many dimes and how many quarters does Lucinda have?

Answer

42 dimes, 8 quarters

4.3-137 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

A cashier has 54 bills, all of which are $10 or $20 bills. The total value of the money is $910. How many of each type of bill does the cashier have?

Answer

17 $10 bills, 37 $20 bills

4.3-139 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

How many pounds of nuts selling for $6 per pound and raisins selling for $3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound?

Answer

80 pounds nuts and 40 pounds raisins

4.3-141 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

Joseph would like to make twelve pounds of a coffee blend at a cost of $6 per pound. He blends Ground Chicory at $5 a pound with Jamaican Blue Mountain at $9 per pound. How much of each type of coffee should he use?

Answer

9 pounds of Chicory coffee, 3 pounds of Jamaican Blue Mountain coffee

4.3-143 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

Twelve-year old Melody wants to sell bags of mixed candy at her lemonade stand. She will mix M&M’s that cost $4.89 per bag and Reese’s Pieces that cost $3.79 per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $4.23 a bag to make. How many bags of M&M’s and how many bags of Reese’s Pieces should she use?

Answer

10 bags of M&M’s, 15 bags of Reese’s Pieces

4.3-145 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

Joy is preparing 15 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?

Answer

7.5

liters of each solution

4.3-147 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

A scientist needs 120 milliliters of a 20% acid solution for an experiment. The lab has available a 25% and a 10% solution. How many liters of the 25% and how many liters of the 10% solutions should the scientist mix to make the 20% solution?

Answer

80 liters of the 25% solution and 40 liters of the 10% solution

4.3-149 §4.3 source ↗

Solve Mixture Applications

In the following exercises, translate to a system of equations and solve.

A 90% antifreeze solution is to be mixed with a 75% antifreeze solution to get 360 liters of an 85% solution. How many liters of the 90% and how many liters of the 75% solutions will be used?

Answer

240 liters of the 90% solution and 120 liters of the 75% solution

4.3-151 §4.3 source ↗

Solve Interest Applications

In the following exercises, translate to a system of equations and solve.

Carol invested $2560 into two accounts. One account paid 8% interest and the other paid 6% interest. She earned $7.25 %$ interest on the total investment. How much money did she put in each account?

Answer

$1600 at 8%, 960 at 6%

4.3-153 §4.3 source ↗

Solve Interest Applications

In the following exercises, translate to a system of equations and solve.

Arnold invested $64,000, some at $5.5 %$ interest and the rest at 9%. How much did he invest at each rate if he received $4500 in interest in one year?

Answer

$28,000 at 9%, $36,000 at

5.5 %

4.3-155 §4.3 source ↗

Solve Interest Applications

In the following exercises, translate to a system of equations and solve.

Mark wants to invest $10,000 to pay for his daughter’s wedding next year. He will invest some of the money in a short term CD that pays 12% interest and the rest in a money market savings account that pays 5% interest. How much should he invest at each rate if he wants to earn $1095 in interest in one year?

Answer

$8500 CD, $1500 savings account

4.3-157 §4.3 source ↗

Solve Interest Applications

In the following exercises, translate to a system of equations and solve.

A business has two loans totaling $85,000. One loan has a rate of 6% and the other has a rate of 4.5% This year, the business expects to pay $4,650 in interest on the two loans. How much is each loan?

Answer

$55,000 on loan at 6% and $30,000 on loan at

4.5 %

4.3-159 §4.3 source ↗

Solve Applications of Cost and Revenue Functions

The manufacturer of a water bottle spends $5 to build each bottle and sells them for $10. The manufacturer also has fixed costs each month of $6500. ⓐ Find the cost function C when x bottles are manufactured. ⓑ Find the revenue function R when x bottles are sold. ⓒ Show the break-even point by graphing both the Revenue and Cost functions on the same grid. ⓓ Find the break-even point. Interpret what the break-even point means.

Answer

ⓐ $C (x) =$ $5 x +$ $6500$

ⓑ $R (x) = 10 x$

ⓒ

Figure shows a graph with two intersecting lines. One of them passes through the origin. The other crosses the y axis at point 6560.

ⓓ 1,300; when 1,300 water bottles are sold, the cost and the revenue equal $13,000

4.3-161 §4.3 source ↗

Solve Applications of Cost and Revenue Functions

In Example 4.28, we used elimination to solve the system of equations
${\begin{matrix} s + b = 40,000 \\ 0.08 s + 0.03 b = 0.071 (40,000) \end{matrix} .$

Could you have used substitution or elimination to solve this system? Why?

Answer

Answers will vary.

Be Prepared 4.10 §4.4 source ↗

Evaluate

5 x - 2 y + 3 z

when

x = −2,

y = −4,

and

z = 3 .

If you missed this problem, review Example 1.21.

Example 1.21 in the book ↗

Answer

7

Be Prepared 4.11 §4.4 source ↗

Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution.

{\begin{matrix} - 2 x + y = −11 \\ x + 3 y = 9 \end{matrix} .

If you missed this problem, review Example 2.6.

Example 2.6 in the book ↗

Answer

conditional; one solution

Be Prepared 4.12 §4.4 source ↗

Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution.

{\begin{matrix} 7 x + 8 y = 4 \\ 3 x - 5 y = 27 \end{matrix} .

If you missed this problem, review Example 2.8.

Example 2.8 in the book ↗

Answer

conditional; one solution

Try It 4.61 §4.4 source ↗

Determine whether the ordered triple is a solution to the system: ${\begin{matrix} 3 x + y + z = 2 \\ x + 2 y + z = −3 \\ 3 x + y + 2 z = 4 \end{matrix} .$

ⓐ $(1, −3, 2)$ ⓑ $(4, −1, −5)$

Answer

ⓐ yes ⓑ no

Try It 4.62 §4.4 source ↗

Determine whether the ordered triple is a solution to the system: ${\begin{matrix} x - 3 y + z = −5 \\ - 3 x - y - z = 1 \\ 2 x - 2 y + 3 z = 1 \end{matrix} .$

ⓐ $(2, −2, 3)$ ⓑ $(−2, 2, 3)$

Answer

ⓐ no ⓑ yes

Try It 4.63 §4.4 source ↗

Solve the system by elimination:

{\begin{matrix} 3 x + y - z = 2 \\ 2 x - 3 y - 2 z = 1 \\ 4 x - y - 3 z = 0 \end{matrix} .

Answer

(2, −1, 3)

Try It 4.64 §4.4 source ↗

Solve the system by elimination:

{\begin{matrix} 4 x + y + z = −1 \\ - 2 x - 2 y + z = 2 \\ 2 x + 3 y - z = 1 \end{matrix} .

Answer

(−2, 3, 4)

Try It 4.65 §4.4 source ↗

Solve:

{\begin{matrix} 3 x - 4 z = −1 \\ 2 y + 3 z = 2 \\ 2 x + 3 y = 6 \end{matrix} .

Answer

(−3, 4, −2)

Try It 4.66 §4.4 source ↗

Solve:

{\begin{matrix} 4 x - 3 z = −5 \\ 3 y + 2 z = 7 \\ 3 x + 4 y = 6 \end{matrix} .

Answer

(−2, 3, −1)

Try It 4.67 §4.4 source ↗

Solve the system of equations:

{\begin{matrix} x + 2 y + 6 z = 5 \\ - x + y - 2 z = 3 \\ x - 4 y - 2 z = 1 \end{matrix} .

Answer

no solution

Try It 4.68 §4.4 source ↗

Solve the system of equations:

{\begin{matrix} 2 x - 2 y + 3 z = 6 \\ 4 x - 3 y + 2 z = 0 \\ - 2 x + 3 y - 7 z = 1 \end{matrix} .

Answer

no solution

Try It 4.69 §4.4 source ↗

Solve the system by equations:

{\begin{matrix} x + y - z = 0 \\ 2 x + 4 y - 2 z = 6 \\ 3 x + 6 y - 3 z = 9 \end{matrix} .

Answer

infinitely many solutions

(x, 3, z)

where

x = z - 3; y = 3; z

is any real number

Try It 4.70 §4.4 source ↗

Solve the system by equations:

{\begin{matrix} x - y - z = 1 \\ - x + 2 y - 3 z = −4 \\ 3 x - 2 y - 7 z = 0 \end{matrix} .

Answer

infinitely many solutions

(x, y, z)

where

x =

5 z -

2;

y =

4 z -

3;

z

is any real number

Try It 4.71 §4.4 source ↗

The community college fine arts department sold three kinds of tickets to its latest dance presentation. The adult tickets sold for $20, the student tickets for $12 and the child tickets for $10.The fine arts department was thrilled to have sold 350 tickets and brought in $4,650 in one night. The number of child tickets sold is the same as the number of adult tickets sold. How many of each type did the department sell?

Answer

The fine arts department sold 75 adult tickets, 200 student tickets, and 75 child tickets.

Try It 4.72 §4.4 source ↗

The community college soccer team sold three kinds of tickets to its latest game. The adult tickets sold for $10, the student tickets for $8 and the child tickets for $5. The soccer team was thrilled to have sold 600 tickets and brought in $4,900 for one game. The number of adult tickets is twice the number of child tickets. How many of each type did the soccer team sell?

Answer

The soccer team sold 200 adult tickets, 300 student tickets, and 100 child tickets.

4.4-163 §4.4 source ↗

Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables

In the following exercises, determine whether the ordered triple is a solution to the system.

${\begin{matrix} - 3 x + y + z = −4 \\ - x + 2 y - 2 z = 1 \\ 2 x - y - z = −1 \end{matrix}$

ⓐ $(−5, −7, 4)$ ⓑ $(5, 7, 4)$

Answer

ⓐ no ⓑ yes

4.4-165 §4.4 source ↗

Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables

In the following exercises, determine whether the ordered triple is a solution to the system.

${\begin{matrix} x + 3 y - z = 15 \\ y = \frac{2}{3} x - 2 \\ x - 3 y + z = −2 \end{matrix}$

ⓐ $(−6, 5, \frac{1}{2})$ ⓑ $(5, \frac{4}{3}, −3)$

Answer

ⓐ no ⓑ no

4.4-167 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} 6 x - 5 y + 2 z = 3 \\ 2 x + y - 4 z = 5 \\ 3 x - 3 y + z = −1 \end{matrix}$

Answer

(4, 5, 2)

4.4-169 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} 5 x - 3 y + 2 z = −5 \\ 2 x - y - z = 4 \\ 3 x - 2 y + 2 z = −7 \end{matrix}$

Answer

(7, 12, −2)

4.4-171 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} 4 x - 3 y + z = 7 \\ 2 x - 5 y - 4 z = 3 \\ 3 x - 2 y - 2 z = −7 \end{matrix}$

Answer

(−3, −5, 4)

4.4-173 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} 11 x + 9 y + 2 z = −9 \\ 7 x + 5 y + 3 z = −7 \\ 4 x + 3 y + z = −3 \end{matrix}$

Answer

(2, −3, −2)

4.4-175 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} x + \frac{1}{2} y + \frac{1}{2} z = 0 \\ \frac{1}{5} x - \frac{1}{5} y + z = 0 \\ \frac{1}{3} x - \frac{1}{3} y + 2 z = −1 \end{matrix}$

Answer

(6, −9, −3)

4.4-177 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} \frac{1}{3} x - y + \frac{1}{2} z = 4 \\ \frac{2}{3} x + \frac{5}{2} y - 4 z = 0 \\ x - \frac{1}{2} y + \frac{3}{2} z = 2 \end{matrix}$

Answer

(3, −4, −2)

4.4-179 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} 2 x + 5 y = 4 \\ 3 y - z = 3 \\ 4 x + 3 z = −3 \end{matrix}$

Answer

(−3, 2, 3)

4.4-181 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} 3 x - z = −3 \\ 5 y + 2 z = −6 \\ 4 x + 3 y = −8 \end{matrix}$

Answer

(−2, 0, −3)

4.4-183 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} x - 2 y + 2 z = 1 \\ - 2 x + y - z = 2 \\ x - y + z = 5 \end{matrix}$

Answer

no solution

4.4-185 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} x + 4 y + z = −8 \\ 4 x - y + 3 z = 9 \\ 2 x + 7 y + z = 0 \end{matrix}$

Answer

x =

\frac{203}{16};

y =

\frac{–25}{16};

z =

\frac{–231}{16};

4.4-187 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} x + y - 2 z = 3 \\ - 2 x - 3 y + z = −7 \\ x + 2 y + z = 4 \end{matrix}$

Answer

(x, y, z)

where

x =

5 z +

2;

y =

−3 z +

1;

z

is any real number

4.4-189 §4.4 source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} x - 2 y + 3 z = 1 \\ x + y - 3 z = 7 \\ 3 x - 4 y + 5 z = 7 \end{matrix}$

Answer

(x, y, z)

where

x =

5 z -

2;

y =

4 z -

3;

z

is any real number

4.4-191 §4.4 source ↗

Solve Applications using Systems of Linear Equations with Three Variables

In the following exercises, solve the given problem.

The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is three times the measure of the first angle. The third angle is fifteen more than the second. Find the measures of the three angles.

Answer

45 degrees, 60 degrees, 75 degrees

4.4-193 §4.4 source ↗

Solve Applications using Systems of Linear Equations with Three Variables

In the following exercises, solve the given problem.

The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of $65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of $140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 cans of popcorn for a total sales of $250. What is the cost of each item?

Answer

$20, $5, $10

4.4-195 §4.4 source ↗

Solve Applications using Systems of Linear Equations with Three Variables

In the following exercises, solve the given problem.

How can you tell when a system of three linear equations with three variables has no solution? Infinitely many solutions?

Answer

Answers will vary.

Be Prepared 4.13 §4.5 source ↗

Solve:

3 (x + 2) +

4 =

4 (2 x - 1) +

9 .

If you missed this problem, review Example 2.2.

Example 2.2 in the book ↗

Answer

x = 1

Be Prepared 4.14 §4.5 source ↗

Solve:

0.25 p +

0.25 (p + 4) =

5.20 .

If you missed this problem, review Example 2.13.

Example 2.13 in the book ↗

Answer

p = 8.4

Be Prepared 4.15 §4.5 source ↗

Evaluate when

x = −2

and

y =

3 : 2 x^{2} -

x y +

3 y^{2} .

If you missed this problem, review Example 1.21.

Example 1.21 in the book ↗

Answer

41

Try It 4.73 §4.5 source ↗

Write each system of linear equations as an augmented matrix:

ⓐ ${\begin{matrix} 3 x + 8 y = −3 \\ 2 x = −5 y - 3 \end{matrix}$ ⓑ ${\begin{matrix} 2 x - 5 y + 3 z = 8 \\ 3 x - y + 4 z = 7 \\ x + 3 y + 2 z = −3 \end{matrix}$

Answer

ⓐ

[\begin{matrix} 3 & 8 & −3 \\ 2 & 5 & −3 \end{matrix}]

ⓑ

[\begin{matrix} 2 & −5 & 3 & 8 \\ 3 & −1 & 4 & 7 \\ 1 & 3 & 2 & −3 \end{matrix}]

Try It 4.74 §4.5 source ↗

Write each system of linear equations as an augmented matrix:

ⓐ ${\begin{matrix} 11 x = −9 y - 5 \\ 7 x + 5 y = −1 \end{matrix}$ ⓑ ${\begin{matrix} 5 x - 3 y + 2 z = −5 \\ 2 x - y - z = 4 \\ 3 x - 2 y + 2 z = −7 \end{matrix}$

Answer

ⓐ

[\begin{matrix} 11 & 9 & −5 \\ 7 & 5 & −1 \end{matrix}]

ⓑ

[\begin{matrix} 5 & −3 & 2 & −5 \\ 2 & −1 & −1 & 4 \\ 3 & −2 & 2 & −7 \end{matrix}]

Try It 4.75 §4.5 source ↗

Write the system of equations that corresponds to the augmented matrix:

[\begin{matrix} 1 & −1 & 2 & 3 \\ 2 & 1 & −2 & 1 \\ 4 & −1 & 2 & 0 \end{matrix}] .

Answer

{\begin{matrix} x - y + 2 z = 3 \\ 2 x + y - 2 z = 1 \\ 4 x - y + 2 z = 0 \end{matrix}

Try It 4.76 §4.5 source ↗

Write the system of equations that corresponds to the augmented matrix:

[\begin{matrix} 1 & 1 & 1 & 4 \\ 2 & 3 & −1 & 8 \\ 1 & 1 & −1 & 3 \end{matrix}] .

Answer

{\begin{matrix} x + y + z = 4 \\ 2 x + 3 y - z = 8 \\ x + y - z = 3 \end{matrix}

Try It 4.77 §4.5 source ↗

Perform the indicated operations sequentially on the augmented matrix:

ⓐ Interchange rows 1 and 3.

ⓑ Multiply row 3 by 3.

$[\begin{matrix} 5 & −2 & −2 \\ 4 & −1 & −4 \\ −2 & 3 & 0 \end{matrix} | \begin{matrix} −2 \\ 4 \\ −1 \end{matrix}]$

Answer

ⓐ

[\begin{matrix} −2 & 3 & 0 & −1 \\ 4 & −1 & −4 & 4 \\ 5 & −2 & −2 & −2 \end{matrix}]

ⓑ

[\begin{matrix} −2 & 3 & 0 & 0 \\ 4 & −1 & −4 & 4 \\ 15 & −6 & −6 & −6 \end{matrix}]

ⓒ

[\begin{matrix} −2 & 3 & 0 & −1 \\ 34 & −13 & −16 & −8 \\ 15 & −6 & −6 & −6 \end{matrix}]

Try It 4.78 §4.5 source ↗

Perform the indicated operations on the augmented matrix:

ⓐ Interchange rows 1 and 2,

ⓑ Multiply row 1 by 2,

$[\begin{matrix} 2 & −3 & −2 \\ 4 & 1 & −3 \\ 5 & 0 & 4 \end{matrix} | \begin{matrix} −4 \\ 2 \\ −1 \end{matrix}]$

Answer

ⓐ

[\begin{matrix} 4 & 1 & −3 & 2 \\ 2 & −3 & −2 & −4 \\ 5 & 0 & 4 & −1 \end{matrix}]

ⓑ

[\begin{matrix} 8 & 2 & −6 & 4 \\ 2 & −3 & −2 & −4 \\ 5 & 0 & 4 & −1 \end{matrix}]

ⓒ

[\begin{matrix} 14 & −7 & −12 & −8 \\ 2 & −3 & −2 & −4 \\ 5 & 0 & 4 & −1 \end{matrix}]

Try It 4.79 §4.5 source ↗

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix:

[\begin{matrix} 1 & −1 \\ 3 & −6 \end{matrix} | \begin{matrix} 2 \\ 2 \end{matrix}] .

Answer

[\begin{matrix} 1 & −1 & 2 \\ 0 & −3 & −4 \end{matrix}]

Try It 4.80 §4.5 source ↗

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix:

[\begin{matrix} 1 & −1 \\ −2 & −3 \end{matrix} | \begin{matrix} 3 \\ 2 \end{matrix}] .

Answer

[\begin{matrix} 1 & −1 & 3 \\ 0 & −5 & 8 \end{matrix}]

Try It 4.81 §4.5 source ↗

Solve the system of equations using a matrix:

{\begin{matrix} 2 x + y = 7 \\ x - 2 y = 6 \end{matrix} .

Answer

The solution is

(4, −1) .

Try It 4.82 §4.5 source ↗

Solve the system of equations using a matrix:

{\begin{matrix} 2 x + y = −4 \\ x - y = −2 \end{matrix} .

Answer

The solution is

(−2, 0) .

Try It 4.83 §4.5 source ↗

Solve the system of equations using a matrix:

{\begin{matrix} 2 x - 5 y + 3 z = 8 \\ 3 x - y + 4 z = 7 \\ x + 3 y + 2 z = −3 \end{matrix} .

Answer

(6, −1, −3)

Try It 4.84 §4.5 source ↗

Solve the system of equations using a matrix:

{\begin{matrix} −3 x + y + z = −4 \\ − x + 2 y - 2 z = 1 \\ 2 x - y - z = −1 \end{matrix} .

Answer

(5, 7, 4)

Try It 4.85 §4.5 source ↗

Solve the system of equations using a matrix:

{\begin{matrix} x - 2 y + 2 z = 1 \\ −2 x + y - z = 2 \\ x - y + z = 5 \end{matrix} .

Answer

no solution

Try It 4.86 §4.5 source ↗

Solve the system of equations using a matrix:

{\begin{matrix} 3 x + 4 y - 3 z = −2 \\ 2 x + 3 y - z = −12 \\ x + y - 2 z = 6 \end{matrix} .

Answer

no solution

Try It 4.87 §4.5 source ↗

Solve the system of equations using a matrix:

{\begin{matrix} x + y - z = 0 \\ 2 x + 4 y - 2 z = 6 \\ 3 x + 6 y - 3 z = 9 \end{matrix} .

Answer

infinitely many solutions

(x, y, z),

where

x = z - 3; y = 3; z

is any real number.

Try It 4.88 §4.5 source ↗

Solve the system of equations using a matrix:

{\begin{matrix} x - y - z = 1 \\ − x + 2 y - 3 z = −4 \\ 3 x - 2 y - 7 z = 0 \end{matrix} .

Answer

infinitely many solutions

(x, y, z),

where

x =

5 z -

2;

y =

4 z -

3;

z

is any real number.

4.5-197 §4.5 source ↗

Write the Augmented Matrix for a System of Equations

In the following exercises, write each system of linear equations as an augmented matrix.

ⓐ ${\begin{matrix} 2 x + 4 y = −5 \\ 3 x - 2 y = 2 \end{matrix}$
ⓑ ${\begin{matrix} 3 x - 2 y - z = −2 \\ −2 x + y = 5 \\ 5 x + 4 y + z = −1 \end{matrix}$

Answer

ⓐ

[\begin{matrix} 2 & 4 & −5 \\ 3 & −2 & 2 \end{matrix}]

ⓑ

[\begin{matrix} 3 & −2 & −1 & −2 \\ −2 & 1 & 0 & 5 \\ 5 & 4 & 1 & −1 \end{matrix}]

4.5-199 §4.5 source ↗

Write the Augmented Matrix for a System of Equations

In the following exercises, write each system of linear equations as an augmented matrix.

ⓐ ${\begin{matrix} 2 x - 5 y = −3 \\ 4 x = 3 y - 1 \end{matrix}$
ⓑ ${\begin{matrix} 4 x + 3 y - 2 z = −3 \\ −2 x + y - 3 z = 4 \\ − x - 4 y + 5 z = −2 \end{matrix}$

Answer

ⓐ

[\begin{matrix} 2 & −5 & −3 \\ 4 & −3 & −1 \end{matrix}]

ⓑ

[\begin{matrix} 4 & 3 & −2 & −3 \\ −2 & 1 & −3 & 4 \\ −1 & −4 & 5 & −2 \end{matrix}]

4.5-201 §4.5 source ↗

Write the system of equations that corresponds to the augmented matrix.

$[\begin{matrix} 2 & −4 \\ 3 & −3 \end{matrix} | \begin{matrix} −2 \\ −1 \end{matrix}]$

Answer

{\begin{matrix} 2 x - 4 y = −2 \\ 3 x - 3 y = −1 \end{matrix}

4.5-203 §4.5 source ↗

Write the system of equations that corresponds to the augmented matrix.

$[\begin{matrix} 2 & −2 & 0 \\ 0 & 2 & −1 \\ 3 & 0 & −1 \end{matrix} | \begin{matrix} −1 \\ 2 \\ −2 \end{matrix}]$

Answer

{\begin{matrix} 2 x - 2 y = −1 \\ 2 y - z = 2 \\ 3 x - z = −2 \end{matrix}

4.5-205 §4.5 source ↗

Use Row Operations on a Matrix

In the following exercises, perform the indicated operations on the augmented matrices.

$[\begin{matrix} 4 & −6 \\ 3 & 2 \end{matrix} | \begin{matrix} −3 \\ 1 \end{matrix}]$

ⓐ Interchange rows 1 and 2

ⓑ Multiply row 1 by 4

Answer

ⓐ

[\begin{matrix} 3 & 2 & 1 \\ 4 & −6 & −3 \end{matrix}]

ⓑ

[\begin{matrix} 12 & 8 & 4 \\ 4 & −6 & −3 \end{matrix}]

ⓒ

[\begin{matrix} 12 & 8 & 4 \\ 24 & −10 & −5 \end{matrix}]

4.5-207 §4.5 source ↗

Use Row Operations on a Matrix

In the following exercises, perform the indicated operations on the augmented matrices.

$[\begin{matrix} 6 & −5 & 2 \\ 2 & 1 & −4 \\ 3 & −3 & 1 \end{matrix} | \begin{matrix} 3 \\ 5 \\ −1 \end{matrix}]$

ⓐ Interchange rows 2 and 3

ⓑ Multiply row 2 by 5

Answer

ⓐ

\begin{matrix} 6 & - 5 & 2 & 3 \\ 3 & - 3 & 1 & - 1 \\ 2 & 1 & - 4 & 5 \end{matrix}

ⓑ

\begin{matrix} 6 & - 5 & 2 & 3 \\ 15 & - 15 & 5 & - 5 \\ 2 & 1 & - 4 & 5 \end{matrix}

ⓒ

\begin{matrix} 2 & - 7 & 10 & - 7 \\ 15 & - 15 & 5 & - 5 \\ 2 & 1 & - 4 & 5 \end{matrix}

4.5-209 §4.5 source ↗

Use Row Operations on a Matrix

In the following exercises, perform the indicated operations on the augmented matrices.

Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix: $[\begin{matrix} 1 & −2 & 3 \\ 3 & −1 & −2 \\ 2 & −3 & −4 \end{matrix} | \begin{matrix} −4 \\ 5 \\ −1 \end{matrix}] .$

Answer

[\begin{matrix} 1 & −2 & 3 & −4 \\ 0 & 5 & −11 & 17 \\ 0 & 1 & −10 & 7 \end{matrix}]

4.5-211 §4.5 source ↗

Solve Systems of Equations Using Matrices

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} 3 x + y = 2 \\ x - y = 2 \end{matrix}$

Answer

(1, −1)

4.5-213 §4.5 source ↗

Solve Systems of Equations Using Matrices

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} −2 x + 3 y = 3 \\ x + 3 y = 12 \end{matrix}$

Answer

(3, 3)

4.5-215 §4.5 source ↗

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} 2 x - y + 3 z = −3 \\ − x + 2 y - z = 10 \\ x + y + z = 5 \end{matrix}$

Answer

(−2, 5, 2)

4.5-217 §4.5 source ↗

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} 4 x - 3 y + z = 7 \\ 2 x - 5 y - 4 z = 3 \\ 3 x - 2 y - 2 z = −7 \end{matrix}$

Answer

(−3, −5, 4)

4.5-219 §4.5 source ↗

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} 2 x + 5 y = 4 \\ 3 y - z = 3 \\ 4 x + 3 z = −3 \end{matrix}$

Answer

(−3, 2, 3)

4.5-221 §4.5 source ↗

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} 3 x - z = −3 \\ 5 y + 2 z = −6 \\ 4 x + 3 y = −8 \end{matrix}$

Answer

(−2, 0, −3)

4.5-223 §4.5 source ↗

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} x + 2 y + 6 z = 5 \\ − x + y - 2 z = 3 \\ x - 4 y - 2 z = 1 \end{matrix}$

Answer

no solution

4.5-225 §4.5 source ↗

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} 4 x - 3 y + 2 z = 0 \\ −2 x + 3 y - 7 z = 1 \\ 2 x - 2 y + 3 z = 6 \end{matrix}$

Answer

no solution

4.5-227 §4.5 source ↗

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} − x - 3 y + 2 z = 14 \\ − x + 2 y - 3 z = −4 \\ 3 x + y - 2 z = 6 \end{matrix}$

Answer

\frac{8}{5}, - \frac{42}{5}, - \frac{24}{5}

4.5-229 §4.5 source ↗

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} x + 2 y + z = 4 \\ x + y - 2 z = 3 \\ −2 x - 3 y + z = −7 \end{matrix}$

Answer

infinitely many solutions

(x, y, z)

where

x =

5 z +

2;

y =

−3 z +

1;

z

is any real number

4.5-231 §4.5 source ↗

In the following exercises, solve each system of equations using a matrix.

Solve the system of equations ${\begin{matrix} 3 x + y = 12 \\ x = y - 8 \end{matrix}$ by substitution and explain all your steps in words.

Answer

Answers will vary.

Be Prepared 4.16 §4.6 source ↗

Simplify:

5 (−2) -

(−4) (1) .

If you missed this problem, review Example 1.20.

Example 1.20 in the book ↗

Answer

- 6

Be Prepared 4.17 §4.6 source ↗

Simplify:

−3 (8 - 10) +

(−2) (6 - 3) -

4 (−3 - (−4)) .

If you missed this problem, review Example 1.19.

Example 1.19 in the book ↗

Answer

- 4

Be Prepared 4.18 §4.6 source ↗

Simplify:

\frac{−12}{−8} .

If you missed this problem, review Example 1.18.

Example 1.18 in the book ↗

Answer

\frac{3}{2}

Try It 4.89 §4.6 source ↗

Evaluate the determinant of ⓐ

[\begin{matrix} 5 −3 \\ 2 −4 \end{matrix}]

ⓑ

[\begin{matrix} −4 & −6 \\ 0 & 7 \end{matrix}] .

Answer

ⓐ

−14;

ⓑ

−28

Try It 4.90 §4.6 source ↗

Evaluate the determinant of ⓐ

[\begin{matrix} −1 3 \\ −2 4 \end{matrix}]

ⓑ

[\begin{matrix} −7 & −3 \\ −5 & 0 \end{matrix}] .

Answer

ⓐ 2 ⓑ

−15

Try It 4.91 §4.6 source ↗

For the determinant

| \begin{matrix} 1 & −1 & 4 \\ 0 & 2 & −1 \\ −2 & −3 & 3 \end{matrix} |,

find and then evaluate the minor of ⓐ

a_{1}

ⓑ

b_{2}

ⓒ

c_{3} .

Answer

ⓐ 3 ⓑ 11 ⓒ 2

Try It 4.92 §4.6 source ↗

For the determinant

| \begin{matrix} −2 & −1 & 0 \\ 3 & 0 & −1 \\ −1 & −2 & 3 \end{matrix} |,

find and then evaluate the minor of ⓐ

a_{2}

ⓑ

b_{3}

ⓒ

c_{2} .

Answer

ⓐ

−3

ⓑ 2 ⓒ 3

Try It 4.93 §4.6 source ↗

Evaluate the determinant

| \begin{matrix} 3 & −2 & 4 \\ 0 & −1 & −2 \\ 2 & 3 & −1 \end{matrix} |,

by expanding by minors along the first row.

Answer

Try It 4.94 §4.6 source ↗

Evaluate the determinant

| \begin{matrix} 3 & −2 & −2 \\ 2 & −1 & 4 \\ −1 & 0 & −3 \end{matrix} |,

by expanding by minors along the first row.

Answer

Try It 4.95 §4.6 source ↗

Evaluate the determinant

| \begin{matrix} 2 & −1 & −3 \\ 0 & 3 & −4 \\ 3 & −4 & −3 \end{matrix} |

by expanding by minors.

Answer

−11

Try It 4.96 §4.6 source ↗

Evaluate the determinant

| \begin{matrix} −2 & −1 & −3 \\ −1 & 2 & 2 \\ 4 & −4 & 0 \end{matrix} |

by expanding by minors.

Answer

−12

Try It 4.97 §4.6 source ↗

Solve using Cramer’s rule:

{\begin{matrix} 3 x + y = −3 \\ 2 x + 3 y = 6 \end{matrix} .

Answer

(- \frac{15}{7}, \frac{24}{7})

Try It 4.98 §4.6 source ↗

Solve using Cramer’s rule:

{\begin{matrix} − x + y = 2 \\ 2 x + y = −4 \end{matrix} .

Answer

(−2, 0)

Try It 4.99 §4.6 source ↗

Solve the system of equations using Cramer’s Rule:

{\begin{matrix} 3 x + 8 y + 2 z = −5 \\ 2 x + 5 y - 3 z = 0 \\ x + 2 y - 2 z = −1 \end{matrix} .

Answer

(−9, 3, −1)

Try It 4.100 §4.6 source ↗

Solve the system of equations using Cramer’s Rule:

{\begin{matrix} 3 x + y - 6 z = −3 \\ 2 x + 6 y + 3 z = 0 \\ 3 x + 2 y - 3 z = −6 \end{matrix} .

Answer

(−6, 3, −2)

Try It 4.101 §4.6 source ↗

Solve the system of equations using Cramer’s rule:

{\begin{matrix} 4 x - 3 y = 8 \\ 8 x - 6 y = 14 \end{matrix} .

Answer

no solution

Try It 4.102 §4.6 source ↗

Solve the system of equations using Cramer’s rule:

{\begin{matrix} x = −3 y + 4 \\ 2 x + 6 y = 8 \end{matrix} .

Answer

infinite solutions

Try It 4.103 §4.6 source ↗

Determine whether the points

(3, −2),

(5, −3),

and

(1, −1)

are collinear.

Answer

yes

Try It 4.104 §4.6 source ↗

Determine whether the points

(−4, −1),

(−6, 2),

and

(−2, −4)

are collinear.

Answer

yes

4.6-233 §4.6 source ↗

Evaluate the Determinant of a 2 × 2 Matrix

In the following exercises, evaluate the determinant of each square matrix.

$[\begin{matrix} −4 8 \\ −3 5 \end{matrix}]$

Answer

4.6-235 §4.6 source ↗

Evaluate the Determinant of a 2 × 2 Matrix

In the following exercises, evaluate the determinant of each square matrix.

$[\begin{matrix} −2 & 0 \\ 7 & −5 \end{matrix}]$

Answer

4.6-237 §4.6 source ↗

Evaluate the Determinant of a 3 × 3 Matrix

In the following exercises, find and then evaluate the indicated minors.

$| \begin{matrix} −1 & −3 & 2 \\ 4 & −2 & −1 \\ −2 & 0 & −3 \end{matrix} |$
Find the minor ⓐ $a_{1}$ ⓑ $b_{1}$ ⓒ $c_{2}$

Answer

ⓐ 6 ⓑ

−14

ⓒ

−6

4.6-239 §4.6 source ↗

Evaluate the Determinant of a 3 × 3 Matrix

In the following exercises, find and then evaluate the indicated minors.

$| \begin{matrix} −2 & −2 & 3 \\ 1 & −3 & 0 \\ −2 & 3 & −2 \end{matrix} |$
Find the minor ⓐ $a_{3}$ ⓑ $b_{3}$ ⓒ $c_{3}$

Answer

ⓐ 9 ⓑ

−3

ⓒ 8

4.6-241 §4.6 source ↗

In the following exercises, evaluate each determinant by expanding by minors along the first row.

$| \begin{matrix} 4 & −1 & −2 \\ −3 & −2 & 1 \\ −2 & −5 & 7 \end{matrix} |$

Answer

−77

4.6-243 §4.6 source ↗

In the following exercises, evaluate each determinant by expanding by minors along the first row.

$| \begin{matrix} 1 & 3 & −2 \\ 5 & −6 & 4 \\ 0 & −2 & −1 \end{matrix} |$

Answer

4.6-245 §4.6 source ↗

In the following exercises, evaluate each determinant by expanding by minors.

$| \begin{matrix} 4 & −1 & 3 \\ 3 & −2 & 2 \\ −1 & 0 & 4 \end{matrix} |$

Answer

−24

4.6-247 §4.6 source ↗

In the following exercises, evaluate each determinant by expanding by minors.

$| \begin{matrix} 2 & −4 & −3 \\ 5 & −1 & −4 \\ 3 & 2 & 0 \end{matrix} |$

Answer

4.6-249 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} x - 2 y = −5 \\ 2 x - 3 y = −4 \end{matrix}$

Answer

(7, 6)

4.6-251 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} 2 x + y = −4 \\ 3 x - 2 y = −6 \end{matrix}$

Answer

(−2, 0)

4.6-253 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} x - 3 y = −9 \\ 2 x + 5 y = 4 \end{matrix}$

Answer

(−3, 2)

4.6-255 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} 3 x + 8 y = −3 \\ 2 x + 5 y = −3 \end{matrix}$

Answer

(−9, 3)

4.6-257 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} 4 x - 3 y + z = 7 \\ 2 x - 5 y - 4 z = 3 \\ 3 x - 2 y - 2 z = −7 \end{matrix}$

Answer

(−3, −5, 4)

4.6-259 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} 11 x + 9 y + 2 z = −9 \\ 7 x + 5 y + 3 z = −7 \\ 4 x + 3 y + z = −3 \end{matrix}$

Answer

(2, −3, −2)

4.6-261 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} 2 x + 5 y = 4 \\ 3 y - z = 3 \\ 4 x + 3 z = −3 \end{matrix}$

Answer

(−3, 2, 3)

4.6-263 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} 3 x - z = −3 \\ 5 y + 2 z = −6 \\ 4 x + 3 y = −8 \end{matrix}$

Answer

(−2, 0, −3)

4.6-265 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} x - 4 y = −1 \\ −3 x + 12 y = 3 \end{matrix}$

Answer

infinitely many solutions

4.6-267 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} 4 x + 3 y = 2 \\ 20 x + 15 y = 5 \end{matrix}$

Answer

inconsistent

4.6-269 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{matrix}$

Answer

inconsistent

4.6-271 §4.6 source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s Rule.

${\begin{matrix} x - 2 y + 3 z = 1 \\ x + y - 3 z = 7 \\ 3 x - 4 y + 5 z = 7 \end{matrix}$

Answer

infinitely many solutions

4.6-273 §4.6 source ↗

Solve Applications Using Determinants

In the following exercises, determine whether the given points are collinear.

$(0, −5),$ $(−2, −2),$ and $(2, −8) .$

Answer

yes

4.6-275 §4.6 source ↗

Solve Applications Using Determinants

In the following exercises, determine whether the given points are collinear.

$(−2, 1),$ $(−4, 4),$ and $(0, −2) .$

Answer

yes

4.6-277 §4.6 source ↗

Solve Applications Using Determinants

In the following exercises, determine whether the given points are collinear.

Explain what is meant by the minor of an entry in a square matrix.

Answer

Answers will vary.

4.6-279 §4.6 source ↗

Solve Applications Using Determinants

In the following exercises, determine whether the given points are collinear.

Explain the steps for solving a system of equations using Cramer’s rule.

Answer

Answers will vary.

Be Prepared 4.19 §4.7 source ↗

Solve the inequality

2 a <

5 a +

12 .

If you missed this problem, review Example 2.52.

Example 2.52 in the book ↗

Answer

- 4 < a

Be Prepared 4.20 §4.7 source ↗

Determine whether the ordered pair

(3, \frac{1}{2})

is a solution to the system

y > 2 x + 3 .

If you missed this problem, review Example 3.34.

Example 3.34 in the book ↗

Answer

Try It 4.105 §4.7 source ↗

Determine whether the ordered pair is a solution to the system: ${\begin{matrix} x - 5 y > 10 \\ 2 x + 3 y > −2 \end{matrix} .$

ⓐ $(3, −1)$ ⓑ $(6, −3)$

Answer

ⓐ no ⓑ yes

Try It 4.106 §4.7 source ↗

Determine whether the ordered pair is a solution to the system: ${\begin{matrix} y > 4 x - 2 \\ 4 x - y < 20 \end{matrix} .$

ⓐ $(−2, 1)$ ⓑ $(4, −1)$

Answer

ⓐ yes ⓑ no

Try It 4.107 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} y < 3 x + 2 \\ y > − x - 1 \end{matrix} .

Answer

The figure shows a graph plotted for the inequalities y less than three times x plus two and y greater than minus x minus one. Two lines intersect each other on the graph. An area to the right of both the lines is colored in grey. It is the solution.

The solution is the grey region.

Try It 4.108 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} y < - \frac{1}{2} x + 3 \\ y < 3 x - 4 \end{matrix} .

Answer

The figure shows the graph plotted for the inequalities y less than minus half of x plus three and y less than three times x minus four. Two intersecting lines are shown on the graph. The area bound by the two lines to the bottom right is shown in grey. It is the solution.

The solution is the grey region.

Try It 4.109 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} x + y \leq 2 \\ y \geq \frac{2}{3} x - 1 \end{matrix} .

Answer

The figure shows the graph for the inequalities x plus y less than or equal to two and y greater than or equal to two by three of x minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.

The solution is the grey region.

Try It 4.110 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} 3 x - 2 y \leq 6 \\ y > − \frac{1}{4} x + 5 \end{matrix} .

Answer

The figure shows graph for the inequalities three times x minus two times y less than or equal to six and y greater than or equal to minus one by four of x plus five. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.

The solution is the grey region.

Try It 4.111 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} y \geq 3 x - 2 \\ y < - 1 \end{matrix} .

Answer

The figure shows graph for the inequalities y greater than or equal to three times x minus two and y less than minus one. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution

The solution is the grey region.

Try It 4.112 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} x > −4 \\ x - 2 y \geq −4 \end{matrix} .

Answer

The figure shows graph for the inequalities x greater than or equal to minus four and x minus two times y greater than minus four. Two intersecting lines are shown and the region bound by both the lines is the marked in grey. It is the solution.

The solution is the grey region.

Try It 4.113 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} 3 x - 2 y \geq 12 \\ y \geq \frac{3}{2} x + 1 \end{matrix} .

Answer

The graph of three times x minus two times y greater than or equal to twelve and y greater than or equal to three by two of x plus one is shown. Two intersecting lines are shown. The inequalities do not have a solution.

No solution.

Try It 4.114 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} x + 3 y > 8 \\ y < - \frac{1}{3} x - 2 \end{matrix} .

Answer

The graph of x plus three times y greater than eight and y less than minus one by three of x minus two is shown. Two intersecting lines are shown. The inequalities do not have a solution.

No solution.

Try It 4.115 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} y \geq 3 x + 1 \\ −3 x + y \geq −4 \end{matrix} .

Answer

The figure shows the graph of the inequalities y greater than or equal to three times x plus one and minus three times x plus y greater than or equal to minus four. Two parallel lines are shown and the region to the left of both is colored in grey. It is the solution.

The solution is the grey region.

Try It 4.116 §4.7 source ↗

Solve the system by graphing:

{\begin{matrix} y \leq - \frac{1}{4} x + 2 \\ x + 4 y \leq 4 \end{matrix} .

Answer

The figure shows the graph of the inequalities y less than or equal to minus one fourth of x plus 2 and x plus four times y less than or equal to four. Two parallel lines are shown and the region to the bottom of both is colored in grey. It is the solution.

The solution is the grey region.

Try It 4.117 §4.7 source ↗

A trailer can carry a maximum weight of 160 pounds and a maximum volume of 15 cubic feet. A microwave oven weighs 30 pounds and has 2 cubic feet of volume, while a printer weighs 20 pounds and has 3 cubic feet of space.

ⓐ Write a system of inequalities to model this situation.
ⓑ Graph the system.
ⓒ Could 4 microwaves and 2 printers be carried on this trailer?
ⓓ Could 7 microwaves and 3 printers be carried on this trailer?

Answer

ⓐ ${\begin{matrix} 30 m + 20 p \leq 160 \\ 2 m + 3 p \leq 15 \end{matrix}$
ⓑ

The graph of two intersecting lines, one red and one blue, is shown. The area bound by the two lines is shown in grey.

ⓒ yes
ⓓ no

Try It 4.118 §4.7 source ↗

Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost $2 and the answer sheets cost $1. Mary’s budget for these supplies allows for a maximum cost of $400.

ⓐ Write a system of inequalities to model this situation.
ⓑ Graph the system.
ⓒ Could Mary purchase 100 pencils and 100 answer sheets?
ⓓ Could Mary purchase 150 pencils and 150 answer sheets?

Answer

ⓐ ${\begin{matrix} a \geq p + 5 \\ a + 2 p \leq 400 \end{matrix}$
ⓑ

ⓒ no
ⓓ no

Try It 4.119 §4.7 source ↗

Tenison needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only $25 to spend on the extra food he needs and will spend it on $0.75 donuts which have 360 calories each and $2 energy drinks which have 110 calories.

ⓐ Write a system of inequalities that models this situation.
ⓑ Graph the system.
ⓒ Can he buy 8 donuts and 4 energy drinks and satisfy his caloric needs?
ⓓ Can he buy 1 donut and 3 energy drinks and satisfy his caloric needs?

Answer

ⓐ ${\begin{matrix} 0.75 d + 2 e \leq 25 \\ 360 d + 110 e \geq 1000 \end{matrix}$
ⓑ

ⓒ yes
ⓓ no

Try It 4.120 §4.7 source ↗

Philip’s doctor tells him he should add at least 1,000 more calories per day to his usual diet. Philip wants to buy protein bars that cost $1.80 each and have 140 calories and juice that costs $1.25 per bottle and have 125 calories. He doesn’t want to spend more than $12.

ⓐ Write a system of inequalities that models this situation.
ⓑ Graph the system.
ⓒ Can he buy 3 protein bars and 5 bottles of juice?
ⓓ Can he buy 5 protein bars and 3 bottles of juice?

Answer

ⓐ ${\begin{matrix} 140 p + 125 j \geq 1000 \\ 1.80 p + 1.25 j \leq 12 \end{matrix}$
ⓑ

ⓒ yes
ⓓ no

4.7-281 §4.7 source ↗

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

In the following exercises, determine whether each ordered pair is a solution to the system.

${\begin{matrix} 4 x - y < 10 \\ −2 x + 2 y > −8 \end{matrix}$

ⓐ $(5, −2)$ ⓑ $(−1, 3)$

Answer

ⓐ false ⓑ true

4.7-283 §4.7 source ↗

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

In the following exercises, determine whether each ordered pair is a solution to the system.

${\begin{matrix} y < \frac{3}{2} x + 3 \\ \frac{3}{4} x - 2 y < 5 \end{matrix}$

ⓐ $(−4, −1)$ ⓑ $(8, 3)$

Answer

ⓐ false ⓑ true

4.7-285 §4.7 source ↗

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

In the following exercises, determine whether each ordered pair is a solution to the system.

${\begin{matrix} 6 x - 5 y < 20 \\ −2 x + 7 y > −8 \end{matrix}$

ⓐ $(1, −3)$ ⓑ $(−4, 4)$

Answer

ⓐ false ⓑ true

4.7-287 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y < - 2 x + 2 \\ y \geq - x - 1 \end{matrix}$

Answer

The figure shows the graph of inequalities y less than minus two times x plus two and y greater than or equal to minus x minus one. Two intersecting lines are shown, one in red and the other in blue. The area bound by the two lines is shown in grey.

The solution is the grey region.

4.7-289 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y \geq - \frac{2}{3} x + 2 \\ y > 2 x - 3 \end{matrix}$

Answer

The figure shows the graph of the inequalities y greater than or equal to minus two by three x plus two and y greater than two times x minus three. Two intersecting lines, one in red and the other in blue, are shown. The region bound by them is shown in grey.

The solution is the grey region.

4.7-291 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} x + 2 y < 4 \\ y < x - 2 \end{matrix}$

Answer

The figure shows the graph of the inequalities x minus two times y less than four and y less than x minus two. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey.

The solution is the grey region.

4.7-293 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} 2 x + 4 y \geq 8 \\ y \leq \frac{3}{4} x \end{matrix}$

Answer

The figure shows the graph of the inequalities two times x plus four times y greater than or equal to eight and y less than or equal to minus three fourth of x. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

4.7-295 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} 3 x - 2 y \leq 6 \\ −4 x - 2 y > 8 \end{matrix}$

Answer

The figure shows the graph of the inequalities three times x minus two times y less than or equal to six and minus four times x minus two times y greater than eight. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

4.7-297 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} 2 x + y > −6 \\ − x + 2 y \geq −4 \end{matrix}$

Answer

The figure shows the graph of the inequalities two times x plus y greater than minus six and minus x plus two times y greater than or equal to minus four. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

4.7-299 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} x - 3 y > 4 \\ y \leq - 1 \end{matrix}$

Answer

The figure shows the graph of the inequalities x minus three times y greater than four and y less than or equal to minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

4.7-301 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y \leq - \frac{2}{3} x + 5 \\ x \geq 3 \end{matrix}$

Answer

The figure shows the graph of the inequality y less than or equal to minus two by three times x plus five and x greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

4.7-303 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y \leq - \frac{1}{2} x + 3 \\ y < 1 \end{matrix}$

Answer

The figure shows the graph of the inequalities y less than or equal to minus half x plus three and y less than one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

4.7-305 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} −3 x + 5 y > 10 \\ x > −1 \end{matrix}$

Answer

The figure shows the graph of the inequalities minus three times x plus five times y greater than ten and x greater than minus one. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

4.7-307 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} x \leq −1 \\ y \geq 3 \end{matrix}$

Answer

The figure shows the graph of the inequalities x less than or equal to minus one and y greater than or equal to three. Two intersecting lines, one in blue and the other in red, are shown. The area bound by the lines is shown in grey. It is the solution.

The solution is the grey region.

4.7-309 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} x - 3 y \geq 6 \\ y > \frac{1}{3} x + 1 \end{matrix}$

Answer

The figure shows the graph of the inequalities x minus three times y greater than or equal to six and y greater than one third of x plus one. Two non intersecting lines, one in blue and the other in red, are shown.

No solution.

4.7-311 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} −3 x + 6 y > 12 \\ 4 y \leq 2 x - 4 \end{matrix}$

Answer

The figure shows the graph of the inequalities minus three times x plus six times y greater than twelve and four times y less than or equal to two times x minus four. Two non intersecting lines, one in blue and the other in red, are shown.

No solution.

4.7-313 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y \geq \frac{1}{2} x - 1 \\ −2 x + 4 y \geq 4 \end{matrix}$

Answer

The figure shows the graph of the inequalities y greater than or equal to minus half x minus one and minus two times x plus four times y greater than or equal to four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.

The solution is the grey region.

4.7-315 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y \geq 3 x - 1 \\ −3 x + y > −4 \end{matrix}$

Answer

The figure shows the graph of the inequalities y greater than or equal to three times x minus one and minus three times x plus y greater than minus four. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.

The solution is the grey region.

4.7-317 §4.7 source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y < \frac{3}{4} x - 2 \\ −3 x + 4 y < 7 \end{matrix}$

Answer

The figure shows the graph of the inequalities y less than three by fourth x minus two and minus three x plus four y less than seven. Two non intersecting lines, one in blue and the other in red, are shown. The solution area is shown in grey.

The solution is the grey region.

4.7-319 §4.7 source ↗

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

Jake does not want to spend more than $50 on bags of fertilizer and peat moss for his garden. Fertilizer costs $2 a bag and peat moss costs $5 a bag. Jake’s van can hold at most 20 bags.

ⓐ Write a system of inequalities to model this situation.
ⓑ Graph the system.
ⓒ Can he buy 15 bags of fertilizer and 4 bags of peat moss?
ⓓ Can he buy 10 bags of fertilizer and 10 bags of peat moss?

Answer

ⓐ ${\begin{matrix} f \geq 0 \\ p \geq 0 \\ f + p \leq 20 \\ 2 f + 5 p \leq 50 \end{matrix}$
ⓑ

The figure shows the graph of the inequalities f plus p less than or equal to twenty and two f and five p less than or equal to fifty. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

ⓒ yes
ⓓ no

4.7-321 §4.7 source ↗

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

Juan is studying for his final exams in chemistry and algebra. he knows he only has 24 hours to study, and it will take him at least three times as long to study for algebra than chemistry.

ⓐ Write a system of inequalities to model this situation.
ⓑ Graph the system.
ⓒ Can he spend 4 hours on chemistry and 20 hours on algebra?
ⓓ Can he spend 6 hours on chemistry and 18 hours on algebra?

Answer

ⓐ ${\begin{matrix} c \geq 0 \\ a \geq 0 \\ c + a \leq 24 \\ a \geq 3 c \end{matrix}$
ⓑ

The figure shows the graph of the inequalities c plus a less than or equal to twenty four and a greater than or equal to three times c. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

ⓒ yes
ⓓ no

4.7-323 §4.7 source ↗

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein a day. A bottle of protein water costs $3.20 and a protein bar costs $1.75. The protein water supplies 27 grams of protein and the bar supplies 16 gram. If he has $10 dollars to spend

ⓐ Write a system of inequalities to model this situation.
ⓑ Graph the system.
ⓒ Could he buy 3 bottles of protein water and 1 protein bar?
ⓓ Could he buy no bottles of protein water and 5 protein bars?

Answer

ⓐ ${\begin{matrix} w \geq 0 \\ b \geq 0 \\ 27 w + 16 b > 80 \\ 3.20 w + 1.75 b \leq 10 \end{matrix}$
ⓑ

The figure shows the graph of the inequalities twenty seven times w plus sixteen times b greater than eighty and three point two times w plus one point seven five b less than or equal to ten. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

ⓒ no
ⓓ yes

4.7-325 §4.7 source ↗

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

Mark is increasing his exercise routine by running and walking at least 4 miles each day. His goal is to burn a minimum of 1500 calories from this exercise. Walking burns 270 calories/mile and running burns 650 calories.

ⓐ Write a system of inequalities to model this situation.
ⓑ Graph the system.
ⓒ Could he meet his goal by walking 3 miles and running 1 mile?
ⓓ Could he meet his goal by walking 2 miles and running 2 miles?

Answer

ⓐ ${\begin{matrix} w \geq 0 \\ r \geq 0 \\ w + r \geq 4 \\ 270 w + 650 r \geq 1500 \end{matrix}$
ⓑ

The figure shows the graph of the inequalities w plus r greater than or equals to four and two seventy w plus six fifty r greater than or equal to fifteen hundred. Two intersecting lines, one in blue and the other in red, are shown. An area is shown in grey.

ⓒ no
ⓓ yes

4.7-327 §4.7 source ↗

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

Graph the system ${\begin{matrix} x + 2 y \leq 6 \\ y \geq - \frac{1}{2} x - 4 \end{matrix} .$ What does the solution mean?

Answer

Answers will vary.

4.R-329 §4.R source ↗

Determine Whether an Ordered Pair is a Solution of a System of Equations.

In the following exercises, determine if the following points are solutions to the given system of equations.

${\begin{matrix} x + y = 8 \\ y = x - 4 \end{matrix}$
ⓐ $(6, 2)$
ⓑ $(9, −1)$

Answer

ⓐ yes ⓑ no

4.R-331 §4.R source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} x + 4 y = −1 \\ x = 3 \end{matrix}$

Answer

The figure shows the graph of equations x plus four times y equal to minus one and x equal to three. Two intersecting lines are shown.

$(3, −1)$

4.R-333 §4.R source ↗

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

${\begin{matrix} − x + 2 y = 4 \\ y = \frac{1}{2} x - 3 \end{matrix}$

Answer

The figure shows the graph for the equations minus x plus two times y equal to four and y equal to half x minus three. Two parallel lines are shown.

no solution

4.R-335 §4.R source ↗

In the following exercises, without graphing determine the number of solutions and then classify the system of equations.

${\begin{matrix} 3 x + 2 y = 6 \\ y = −3 x + 4 \end{matrix}$

Answer

one solution, consistent system, independent equations

4.R-337 §4.R source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} 3 x - 2 y = 2 \\ y = \frac{1}{2} x + 3 \end{matrix}$

Answer

(4, 5)

4.R-339 §4.R source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} y = −2 x + 7 \\ y = \frac{2}{3} x - 1 \end{matrix}$

Answer

(3, 1)

4.R-341 §4.R source ↗

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

${\begin{matrix} y = - \frac{1}{3} x + 2 \\ x + 3 y = 6 \end{matrix}$

Answer

infinitely many solutions

4.R-343 §4.R source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination

${\begin{matrix} 3 x - 8 y = 20 \\ x + 3 y = 1 \end{matrix}$

Answer

(4, −1)

4.R-345 §4.R source ↗

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination

${\begin{matrix} \frac{1}{3} x - \frac{1}{2} y = 1 \\ \frac{3}{4} x - y = \frac{5}{2} \end{matrix}$

Answer

(6, 2)

4.R-347 §4.R source ↗

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

${\begin{matrix} 6 x - 5 y = 27 \\ 3 x + 10 y = −24 \end{matrix}$

Answer

elimination

4.R-349 §4.R source ↗

Solve Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Mollie wants to plant 200 bulbs in her garden, all irises and tulips. She wants to plant three times as many tulips as irises. How many irises and how many tulips should she plant?

Answer

50 irises and 150 tulips

4.R-351 §4.R source ↗

Solve Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Leroy spent 20 minutes jogging and 40 minutes cycling and burned 600 calories. The next day, Leroy swapped times, doing 40 minutes of jogging and 20 minutes of cycling and burned the same number of calories. How many calories were burned for each minute of jogging and how many for each minute of cycling?

Answer

10 calories jogging and 10 calories cycling

4.R-353 §4.R source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

The difference of two supplementary angles is 58 degrees. Find the measures of the angles.

Answer

119 degrees and 61 degrees

4.R-355 §4.R source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

Answer

35 degrees and 55 degrees

4.R-357 §4.R source ↗

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

The perimeter of a city rectangular park is 1428 feet. The length is 78 feet more than twice the width. Find the length and width of the park.

Answer

Length = 502 feet, Width = 212 feet

4.R-359 §4.R source ↗

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

Bob left home, riding his bike at a rate of 10 miles per hour to go to the lake. Cheryl, his wife, left 45 minutes $(\frac{3}{4}$ hour) later, driving her car at a rate of 25 miles per hour. How long will it take Cheryl to catch up to Bob?

Answer

\frac{1}{2}

an hour

4.R-361 §4.R source ↗

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

A passenger jet can fly 804 miles in 2 hours with a tailwind but only 776 miles in 2 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Answer

the rate of the jet is 395 mph, the rate of the wind is 7 mph

4.R-363 §4.R source ↗

Solve Mixture Applications with Systems of Equations

For the following exercises, translate to a system of equations and solve.

Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?

Answer

41 dimes and 11 pennies

4.R-365 §4.R source ↗

Solve Mixture Applications with Systems of Equations

For the following exercises, translate to a system of equations and solve.

A scientist needs 70 liters of a 40% solution of alcohol. He has a 30% and a 60% solution available. How many liters of the 30% and how many liters of the 60% solutions should he mix to make the 40% solution?

Answer

46 \frac{2}{3}

liters of 30% solution,

23 \frac{1}{3}

liters of 60% solution

4.R-367 §4.R source ↗

Solve Interest Applications

For the following exercises, translate to a system of equations and solve.

When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?

Answer

$29,000 for the federal loan, $14,000 for the private loan

4.R-369 §4.R source ↗

Solve Systems of Equations with Three Variables

In the following exercises, determine whether the ordered triple is a solution to the system.

${\begin{matrix} y = \frac{2}{3} x - 2 \\ x + 3 y - z = 15 \\ x - 3 y + z = −2 \end{matrix}$
ⓐ $(−6, 5, \frac{1}{2})$
ⓑ $(5, \frac{4}{3}, −3)$

Answer

ⓐ no ⓑ no

4.R-371 §4.R source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} x + \frac{5}{2} y + z = −2 \\ 2 x + 2 y + \frac{1}{2} z = −4 \\ \frac{1}{3} x - y - z = 1 \end{matrix}$

Answer

(−3, 2, −4)

4.R-373 §4.R source ↗

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

${\begin{matrix} 2 x + 3 y + z = 12 \\ x + y + z = 9 \\ 3 x + 4 y + 2 z = 20 \end{matrix}$

Answer

no solution

4.R-375 §4.R source ↗

Solve Applications using Systems of Linear Equations with Three Variables

After attending a major league baseball game, the patrons often purchase souvenirs. If a family purchases 4 t-shirts, a cap and 1 stuffed animal their total is $135. A couple buys 2 t-shirts, a cap and 3 stuffed animals for their nieces and spends $115. Another couple buys 2 t-shirts, a cap and 1 stuffed animal and their total is $85. What is the cost of each item?

Answer

25, 20, 15

4.R-377 §4.R source ↗

Write the Augmented Matrix for a System of Equations.

Write each system of linear equations as an augmented matrix.

${\begin{matrix} 4 x + 3 y = −2 \\ x - 2 y - 3 z = 7 \\ 2 x - y + 2 z = −6 \end{matrix}$

Answer

[\begin{matrix} 4 & 3 & 0 & −2 \\ 1 & −2 & −3 & 7 \\ 2 & −1 & 2 & −6 \end{matrix}]

4.R-379 §4.R source ↗

Write the system of equations that that corresponds to the augmented matrix.

$[\begin{matrix} 1 & 0 & −3 \\ 1 & −2 & 0 \\ 0 & −1 & 2 \end{matrix} | \begin{matrix} −1 \\ −2 \\ 3 \end{matrix}]$

Answer

{\begin{matrix} x - 3 z = −1 \\ x - 2 y = −2 \\ − y + 2 z = 3 \end{matrix}

4.R-381 §4.R source ↗

In the following exercises, perform the indicated operations on the augmented matrices.

$[\begin{matrix} 1 & −3 & −2 \\ 2 & 2 & −1 \\ 4 & −2 & −3 \end{matrix} | \begin{matrix} 4 \\ −3 \\ −1 \end{matrix}]$

Answer

ⓐ

[\begin{matrix} 1 & −3 & −2 & 4 \\ 4 & −2 & −3 & −1 \\ 2 & 2 & −1 & −3 \end{matrix}]

ⓑ

[\begin{matrix} 2 & −6 & −4 & 8 \\ 4 & −2 & −3 & −1 \\ 2 & 2 & −1 & −3 \end{matrix}]

ⓒ

[\begin{matrix} 2 & −6 & −4 & 8 \\ 4 & −2 & −3 & −1 \\ 0 & −6 & −1 & 5 \end{matrix}]

4.R-383 §4.R source ↗

Solve Systems of Equations Using Matrices

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} 2 x - y + 3 z = −3 \\ − x + 2 y - z = 10 \\ x + y + z = 5 \end{matrix}$

Answer

(−2, 5, 2)

4.R-385 §4.R source ↗

Solve Systems of Equations Using Matrices

In the following exercises, solve each system of equations using a matrix.

${\begin{matrix} x + 2 y - 3 z = −1 \\ x - 3 y + z = 1 \\ 2 x - y - 2 z = 2 \end{matrix}$

Answer

no solution

4.R-387 §4.R source ↗

Evaluate the Determinant of a 2 × 2 Matrix

In the following exercise, evaluate the determinate of the square matrix.

$[\begin{matrix} 8 & −4 \\ 5 & −3 \end{matrix}]$

Answer

−4

4.R-389 §4.R source ↗

In the following exercise, evaluate each determinant by expanding by minors along the first row.

$| \begin{matrix} −2 & −3 & −4 \\ 5 & −6 & 7 \\ −1 & 2 & 0 \end{matrix} |$

Answer

4.R-391 §4.R source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s rule

${\begin{matrix} x - 3 y = −9 \\ 2 x + 5 y = 4 \end{matrix}$

Answer

(−3, 2)

4.R-393 §4.R source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s rule

${\begin{matrix} 2 x + 5 y = 4 \\ 3 y - z = 3 \\ 4 x + 3 z = −3 \end{matrix}$

Answer

(−3, 2, 3)

4.R-395 §4.R source ↗

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s rule

${\begin{matrix} 3 x + 4 y - 3 z = −2 \\ 2 x + 3 y - z = −12 \\ x + y - 2 z = 6 \end{matrix}$

Answer

inconsistent

4.R-397 §4.R source ↗

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

In the following exercises, determine whether each ordered pair is a solution to the system.

${\begin{matrix} 4 x + y > 6 \\ 3 x - y \leq 12 \end{matrix}$

ⓐ $(2, −1)$
ⓑ $(3, −2)$

Answer

ⓐ yes ⓑ yes

4.R-399 §4.R source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} y \leq 3 x + 2 \\ y > - x - 1 \end{matrix}$

Answer

The figure shows the graph of inequalities y less than three times x plus one and y greater than or equal to minus x minus two. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.

The solution is the grey region.

4.R-401 §4.R source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} 2 x - 3 y < 6 \\ 3 x + 4 y \geq 12 \end{matrix}$

Answer

The figure shows the graph of inequalities two times x minus three times y less six and three times x plus four times y greater than or equal to twelve. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.

The solution is the grey region.

4.R-403 §4.R source ↗

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

${\begin{matrix} x + 3 y < 5 \\ y \geq - \frac{1}{3} x + 6 \end{matrix}$

Answer

The figure shows the graph of inequalities x plus three times y less than five and y greater than or equal to minus one third x plus six. Two parallel lines, one in red and the other in blue, are shown. An area is shown in grey.

No solution.

4.R-405 §4.R source ↗

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

Roxana makes bracelets and necklaces and sells them at the farmers’ market. She sells the bracelets for $12 each and the necklaces for $18 each. At the market next weekend she will have room to display no more than 40 pieces, and she needs to sell at least $500 worth in order to earn a profit.

ⓐ Write a system of inequalities to model this situation.
ⓑ Graph the system.
ⓒ Should she display 26 bracelets and 14 necklaces?
ⓓ Should she display 39 bracelets and 1 necklace?

Answer

ⓐ ${\begin{matrix} b \geq 0 \\ n \geq 0 \\ b + n \leq 40 \\ 12 b + 18 n \geq 500 \end{matrix}$
ⓑ

The figure shows the graph of b plus n equal to forty and twelve b plus eighteen n equal to five hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.

ⓒ yes
ⓓ no

4.T-407 §4.T source ↗

In the following exercises, solve the following systems by graphing.

${\begin{matrix} x - y = 5 \\ x + 2 y = −4 \end{matrix}$

Answer

The figure shows the graph of inequalities h equal to three p plus five and four times p plus fifteen times h equal to six hundred. Two intersecting lines, one in red and the other in blue, are shown. An area is shown in grey.

$(2, −3)$

4.T-409 §4.T source ↗

In the following exercises, solve each system of equations. Use either substitution or elimination.

${\begin{matrix} x + 4 y = 6 \\ −2 x + y = −3 \end{matrix}$

Answer

(2, 1)

4.T-411 §4.T source ↗

In the following exercises, solve each system of equations. Use either substitution or elimination.

${\begin{matrix} x + y - z = −1 \\ 2 x - y + 2 z = 8 \\ −3 x + 2 y + z = −9 \end{matrix}$

Answer

(2, −2, 1)

4.T-413 §4.T source ↗

Solve the system of equations using a matrix.

${\begin{matrix} −3 x + y + z = −4 \\ − x + 2 y - 2 z = 1 \\ 2 x - y - z = −1 \end{matrix}$

Answer

(5, 7, 4)

4.T-415 §4.T source ↗

Solve using Cramer’s rule.

Evaluate the determinant by expanding by minors:
$| \begin{matrix} 3 & −2 & −2 \\ 2 & −1 & 4 \\ −1 & 0 & −3 \end{matrix} | .$

Answer

7

4.T-417 §4.T source ↗

In the following exercises, translate to a system of equations and solve.

A pharmacist needs 20 liters of a 2% saline solution. He has a 1% and a 5% solution available. How many liters of the 1% and how many liters of the 5% solutions should she mix to make the 2% solution?

Answer

15 liters of 1% solution, 5 liters of 5% solution

4.T-419 §4.T source ↗

In the following exercises, translate to a system of equations and solve.

The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of $65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of $140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 can of popcorn for a total sales of $250. What is the cost of each item?

Answer

The candy cost $20; the cookies cost $5; and the popcorn cost $10.

4.T-421 §4.T source ↗

In the following exercises, translate to a system of equations and solve.

Translate to a system of inequalities and solve.

Andi wants to spend no more than $50 on Halloween treats. She wants to buy candy bars that cost $1 each and lollipops that cost $0.50 each, and she wants the number of lollipops to be at least three times the number of candy bars.

ⓐ Write a system of inequalities to model this situation.
ⓑ Graph the system.
ⓒ Can she buy 20 candy bars and 40 lollipops?

Answer

ⓐ ${\begin{matrix} C \geq 0 \\ L \geq 0 \\ C + 0.5 L \leq 50 \\ L \geq 3 C \end{matrix}$
ⓑ

The figure shows the graph of two equations. Two intersecting lines, one in red and the other in blue, are shown. The red line passes through origin. An area is shown in grey.

ⓒ no
ⓓ yes

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Problems and answers are adapted from Intermediate Algebra 2e by Lynn Marecek and Andrea Honeycutt Mathis, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.