Chapter 5: Systems of Linear EquationsContent check — Elementary Algebra, Chapter 5: Systems of Linear Equations

For the equation $y=\frac{2}{3}x-4$
ⓐ is $\left(6,0\right)$ a solution? ⓑ is $\left(\mathrm{-3},\mathrm{-2}\right)$ a solution?
If you missed this problem, review Example 2.1.

Find the slope and y-intercept of the line $3x-y=12$.
If you missed this problem, review Example 4.42.

Find the x- and y-intercepts of the line $2x-3y=12$.
If you missed this problem, review Example 4.21.

Determine whether the ordered pair is a solution to the system: $$\{\begin{array}{c}3x+y=0\hfill \\ x+2y=\mathrm{-5}\hfill \end{array}.$$

ⓐ $(1,\mathrm{-3})$ ⓑ $(0,0)$

Determine whether the ordered pair is a solution to the system: $$\{\begin{array}{c}x-3y=\mathrm{-8}\hfill \\ \mathrm{-3}x-y=4\hfill \end{array}.$$

ⓐ $(2,\mathrm{-2})$ ⓑ $(\mathrm{-2},2)$

Solve the system by graphing: $$\{\begin{array}{c}x-3y=\mathrm{-3}\hfill \\ x+y=5\hfill \end{array}.$$

Solve the system by graphing: $$\{\begin{array}{c}-x+y=1\hfill \\ 3x+2y=12\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}y=2x+2\hfill \\ y=\text{−}x-4\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}y=3x+3\hfill \\ y=\text{−}x+7\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}-x+y=1\hfill \\ 2x+y=10\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}2x+y=6\hfill \\ x+y=1\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}x+y=6\hfill \\ x-y=2\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}x+y=2\hfill \\ x-y=\mathrm{-8}\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}y=\mathrm{-1}\hfill \\ x+3y=6\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}x=4\hfill \\ 3x-2y=24\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}y=-\frac{1}{4}x+2\hfill \\ x+4y=-8\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}y=3x-1\hfill \\ 6x-2y=6\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}y=-3x-6\hfill \\ 6x+2y=-12\hfill \end{array}.$$

Solve each system by graphing: $$\{\begin{array}{c}y=\frac{1}{2}x-4\hfill \\ 2x-4y=16\hfill \end{array}.$$

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

$$\{\begin{array}{c}y=-2x-4\hfill \\ 4x+2y=9\hfill \end{array}$$

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

$$\{\begin{array}{c}y=\frac{1}{3}x-5\hfill \\ x-3y=6\hfill \end{array}$$

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

$$\{\begin{array}{c}3x+2y=2\hfill \\ 2x+y=1\hfill \end{array}$$

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

$$\{\begin{array}{c}x+4y=12\hfill \\ -x+y=3\hfill \end{array}$$

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

$$\{\begin{array}{c}4x-5y=20\hfill \\ y=\frac{4}{5}x-4\hfill \end{array}$$

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

$$\{\begin{array}{c}\mathrm{-2}x-4y=8\hfill \\ y=-\frac{1}{2}x-2\hfill \end{array}$$

Manny is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manny need?

Alisha is making an 18 ounce coffee beverage that is made from brewed coffee and milk. The number of ounces of brewed coffee is 5 times greater than the number of ounces of milk. How many ounces of coffee and how many ounces of milk does Alisha need?

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.

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

ⓐ $(3,1)$ ⓑ $(\mathrm{-3},4)$

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.

$$\{\begin{array}{c}2x+y=5\hfill \\ x+y=1\hfill \end{array}$$

ⓐ $(4,\text{−3})$ ⓑ $(2,0)$

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.

$$\{\begin{array}{c}x+y=2\hfill \\ y=\frac{3}{4}x\hfill \end{array}$$

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

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.

$$\{\begin{array}{c}x+5y=10\hfill \\ y=\frac{3}{5}x+1\hfill \end{array}$$

ⓐ $\left(\mathrm{-10},4\right)$ ⓑ $\left(\frac{5}{4},\frac{7}{4}\right)$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}3x+y=\mathrm{-3}\hfill \\ 2x+3y=5\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}\mathrm{-3}x+y=\mathrm{-1}\hfill \\ 2x+y=4\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}y=x+2\hfill \\ y=\mathrm{-2}x+2\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

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

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}-x+y=\mathrm{-3}\hfill \\ 4x+4y=4\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}\mathrm{-3}x+y=\mathrm{-1}\hfill \\ 2x+y=4\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}x+y=5\hfill \\ 2x-y=4\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}x+y=2\hfill \\ x-y=0\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}x+y=\mathrm{-5}\hfill \\ x-y=3\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}x+y=\mathrm{-4}\hfill \\ -x+2y=\mathrm{-2}\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}\mathrm{-2}x+3y=3\hfill \\ x+3y=12\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}2x+3y=6\hfill \\ y=\mathrm{-2}\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}x-3y=\mathrm{-3}\hfill \\ y=2\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}2x-y=\mathrm{-1}\hfill \\ x=1\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}x-3y=\mathrm{-6}\hfill \\ x=\mathrm{-3}\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}4x-3y=8\hfill \\ 8x-6y=14\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}\mathrm{-2}x+4y=4\hfill \\ y=\frac{1}{2}x\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}x=\mathrm{-3}y+4\hfill \\ 2x+6y=8\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}2x+y=6\hfill \\ \mathrm{-8}x-4y=\mathrm{-24}\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}x+3y=\mathrm{-6}\hfill \\ 4y=-\frac{4}{3}x-8\hfill \end{array}$$

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

$$\{\begin{array}{c}\mathrm{-3}x+2y=\mathrm{-2}\hfill \\ y=\text{−}x+4\hfill \end{array}$$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$$\{\begin{array}{c}y=\frac{2}{3}x+1\hfill \\ \mathrm{-2}x+3y=5\hfill \end{array}$$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$$\{\begin{array}{c}y=\mathrm{-2}x+1\hfill \\ 4x+2y=8\hfill \end{array}$$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$$\{\begin{array}{c}y=\frac{2}{3}x+1\hfill \\ 2x-3y=7\hfill \end{array}$$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$$\{\begin{array}{c}4x+2y=10\hfill \\ 4x-2y=\mathrm{-6}\hfill \end{array}$$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$$\{\begin{array}{c}y=-\frac{1}{2}x+5\hfill \\ x+2y=10\hfill \end{array}$$

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

$$\{\begin{array}{c}y=2x+3\hfill \\ 2x-y=\mathrm{-3}\hfill \end{array}$$

Solve Applications of Systems of Equations by Graphing In the following exercises, solve.

Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water. How many ounces of strawberry juice and how many ounces of water does she need to make 64 ounces of strawberry infused water?

Solve Applications of Systems of Equations by Graphing In the following exercises, solve.

Enrique is making a party mix that contains raisins and nuts. For each ounce of nuts, he uses twice the amount of raisins. How many ounces of nuts and how many ounces of raisins does he need to make 24 ounces of party mix?

Solve Applications of Systems of Equations by Graphing In the following exercises, solve.

Leo is planning his spring flower garden. He wants to plant tulip and daffodil bulbs. He will plant 6 times as many daffodil bulbs as tulip bulbs. If he wants to plant 350 bulbs, how many tulip bulbs and how many daffodil bulbs should he plant?

Solve Applications of Systems of Equations by Graphing In the following exercises, solve.

In a system of linear equations, the two equations have the same slope. Describe the possible solutions to the system.

Simplify $\mathrm{-5}\left(3-x\right)$.
If you missed this problem, review Example 1.136.

Simplify $4-2\left(n+5\right)$.
If you missed this problem, review Example 1.123.

Solve for $y$: $8y-8=32-2y$
If you missed this problem, review Example 2.34.

Solve for $x$: $3x-9y=\mathrm{-3}$
If you missed this problem, review Example 2.65.

Solve the system by substitution. $$\{\begin{array}{c}\mathrm{-2}x+y=\mathrm{-11}\hfill \\ x+3y=9\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}x+3y=10\hfill \\ 4x+y=18\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}x+y=6\hfill \\ y=3x-2\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}2x-y=1\hfill \\ y=\mathrm{-3}x-6\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}4x+y=2\hfill \\ 3x+2y=\mathrm{-1}\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}-x+y=4\hfill \\ 4x-y=2\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}x-5y=13\hfill \\ 4x-3y=1\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}x-6y=\mathrm{-6}\hfill \\ 2x-4y=4\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}y=3x-16\hfill \\ y=\frac{1}{3}x\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}y=\text{−}x+10\hfill \\ y=\frac{1}{4}x\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}x-4y=\mathrm{-4}\hfill \\ \mathrm{-3}x+4y=0\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}4x-y=0\hfill \\ 2x-3y=5\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}2x-3y=12\hfill \\ \mathrm{-12}y+8x=48\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}5x+2y=12\hfill \\ \mathrm{-4}y-10x=\mathrm{-24}\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}3x+2y=9\hfill \\ y=-\frac{3}{2}x+1\hfill \end{array}$$

Solve the system by substitution. $$\{\begin{array}{c}5x-3y=2\hfill \\ y=\frac{5}{3}x-4\hfill \end{array}$$

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

The sum of two number is −6. One number is 10 less than the other. Find the numbers.

The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle.

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.

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.

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?

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?

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}2x+y=\mathrm{-4}\hfill \\ 3x-2y=\mathrm{-6}\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}x-2y=\mathrm{-5}\hfill \\ 2x-3y=\mathrm{-4}\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}5x-2y=\mathrm{-6}\hfill \\ y=3x+3\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}2x+3y=3\hfill \\ y=\text{−}x+3\hfill \end{array}$$

Solve a System of Equations by Substitution

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

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

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}3x-2y=6\hfill \\ y=\frac{2}{3}x+2\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}2x+y=10\hfill \\ -x+y=\mathrm{-5}\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}3x+y=1\hfill \\ \mathrm{-4}x+y=15\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}x+3y=1\hfill \\ 3x+5y=\mathrm{-5}\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}2x+y=5\hfill \\ x-2y=\mathrm{-15}\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}y=\mathrm{-2}x-1\hfill \\ y=-\frac{1}{3}x+4\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}y=2x-8\hfill \\ y=\frac{3}{5}x+6\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}4x+2y=8\hfill \\ 8x-y=1\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}15x+2y=6\hfill \\ \mathrm{-5}x+2y=\mathrm{-4}\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}y=3x\hfill \\ 6x-2y=0\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}2x+16y=8\hfill \\ -x-8y=\mathrm{-4}\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}y=\mathrm{-4}x\hfill \\ 4x+y=1\hfill \end{array}$$

Solve a System of Equations by Substitution

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

$$\{\begin{array}{c}y=\frac{7}{8}x+4\hfill \\ \mathrm{-7}x+8y=6\hfill \end{array}$$

Solve Applications of Systems of Equations by Substitution

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

The sum of two numbers is 15. One number is 3 less than the other. Find the numbers.

Solve Applications of Systems of Equations by Substitution

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

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

Solve Applications of Systems of Equations by Substitution

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

The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.

Solve Applications of Systems of Equations by Substitution

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

The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.

Solve Applications of Systems of Equations by Substitution

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.

Solve Applications of Systems of Equations by Substitution

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.

Solve Applications of Systems of Equations by Substitution

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?

Solve Applications of Systems of Equations by Substitution

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?

Solve Applications of Systems of Equations by Substitution

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

Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system $$\{\begin{array}{c}56s=70t\hfill \\ s=t+\frac{1}{2}\hfill \end{array}$$.

ⓐ for $t$ to find out how long it will take Tina to catch up to Stephanie.

ⓑ what is the value of $s$, the number of hours Stephanie will have driven before Tina catches up to her?

Solve Applications of Systems of Equations by Substitution

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

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

Simplify $\mathrm{-5}\left(6-3a\right)$.
If you missed this problem, review Example 1.136.

Solve the equation $\frac{1}{3}x+\frac{5}{8}=\frac{31}{24}$.
If you missed this problem, review Example 2.48.

Solve the system by elimination. $$\{\begin{array}{c}3x+y=5\hfill \\ 2x-3y=7\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}4x+y=\mathrm{-5}\hfill \\ \mathrm{-2}x-2y=\mathrm{-2}\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}2x+y=5\hfill \\ x-y=4\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}x+y=3\hfill \\ \mathrm{-2}x-y=\mathrm{-1}\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}4x-3y=1\hfill \\ 5x-9y=\mathrm{-4}\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}3x+2y=2\hfill \\ 6x+5y=8\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}3x-4y=\mathrm{-9}\hfill \\ 5x+3y=14\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}7x+8y=4\hfill \\ 3x-5y=27\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}\frac{1}{3}x-\frac{1}{2}y=1\hfill \\ \frac{3}{4}x-y=\frac{5}{2}\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}x+\frac{3}{5}y=-\frac{1}{5}\hfill \\ -\frac{1}{2}x-\frac{2}{3}y=\frac{5}{6}\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}5x-3y=15\hfill \\ y=\mathrm{-5}+\frac{5}{3}x\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}x+2y=6\hfill \\ y=-\frac{1}{2}x+3\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}\mathrm{-3}x+2y=8\hfill \\ 9x-6y=13\hfill \end{array}$$

Solve the system by elimination. $$\{\begin{array}{c}7x-3y=-2\hfill \\ \mathrm{-14}x+6y=8\hfill \end{array}$$

The sum of two numbers is 42. Their difference is 8. Find the numbers.

The sum of two numbers is −15. Their difference is −35. Find the numbers.

Malik stops at the grocery store to buy a bag of diapers and 2 cans of formula. He spends a total of $37. The next week he stops and buys 2 bags of diapers and 5 cans of formula for a total of $87. How much does a bag of diapers cost? How much is one can of formula?

To get her daily intake of fruit for the day, Sasha eats a banana and 8 strawberries on Wednesday for a calorie count of 145. On the following Wednesday, she eats two bananas and 5 strawberries for a total of 235 calories for the fruit. How many calories are there in a banana? How many calories are in a strawberry?

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

ⓐ $$\{\begin{array}{c}4x-5y=\mathrm{-32}\hfill \\ 3x+2y=\mathrm{-1}\hfill \end{array}$$ ⓑ $$\{\begin{array}{c}x=2y-1\hfill \\ 3x-5y=-7\hfill \end{array}$$

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

ⓐ $$\{\begin{array}{c}y=2x-1\hfill \\ 3x-4y=-6\hfill \end{array}$$ ⓑ $$\{\begin{array}{c}6x-2y=12\hfill \\ 3x+7y=\mathrm{-13}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}\mathrm{-3}x+y=\mathrm{-9}\hfill \\ x-2y=\mathrm{-12}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}3x-y=\mathrm{-7}\hfill \\ 4x+2y=\mathrm{-6}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}x+y=\mathrm{-8}\hfill \\ x-y=\mathrm{-6}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}\mathrm{-7}x+6y=\mathrm{-10}\hfill \\ x-6y=22\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}5x+2y=1\hfill \\ \mathrm{-5}x-4y=\mathrm{-7}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}3x-4y=\mathrm{-11}\hfill \\ x-2y=\mathrm{-5}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}6x-5y=\mathrm{-75}\hfill \\ -x-2y=\mathrm{-13}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}2x-5y=7\hfill \\ 3x-y=17\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}7x+y=\mathrm{-4}\hfill \\ 13x+3y=4\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}3x-5y=\mathrm{-9}\hfill \\ 5x+2y=16\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}4x+7y=14\hfill \\ \mathrm{-2}x+3y=32\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}3x+8y=\mathrm{-3}\hfill \\ 2x+5y=\mathrm{-3}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}3x+8y=67\hfill \\ 5x+3y=60\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}\frac{1}{3}x-y=\mathrm{-3}\hfill \\ x+\frac{5}{2}y=2\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}x+\frac{1}{3}y=\mathrm{-1}\hfill \\ \frac{1}{2}x-\frac{1}{3}y=\mathrm{-2}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}2x+y=3\hfill \\ 6x+3y=9\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}\mathrm{-3}x-y=8\hfill \\ 6x+2y=\mathrm{-16}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}3x+2y=6\hfill \\ \mathrm{-6}x-4y=\mathrm{-12}\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}\mathrm{-11}x+12y=60\hfill \\ \mathrm{-22}x+24y=90\hfill \end{array}$$

Solve a System of Equations by Elimination

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

$$\{\begin{array}{c}5x-3y=15\hfill \\ y=\frac{5}{3}x-2\hfill \end{array}$$

Solve Applications of Systems of Equations by Elimination

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

The sum of two numbers is 65. Their difference is 25. Find the numbers.

Solve Applications of Systems of Equations by Elimination

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

The sum of two numbers is −27. Their difference is −59. Find the numbers.

Solve Applications of Systems of Equations by Elimination

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

Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for $114 or she is able to buy 2 shirts and 4 sweaters for $164. How much does a shirt cost? How much does a sweater cost?

Solve Applications of Systems of Equations by Elimination

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

The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is 4720 mg. The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is 6300 mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?

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.

ⓐ $$\{\begin{array}{c}8x-15y=\mathrm{-32}\hfill \\ 6x+3y=\mathrm{-5}\hfill \end{array}$$ ⓑ $$\{\begin{array}{c}x=4y-3\hfill \\ 4x-2y=\mathrm{-6}\hfill \end{array}$$

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.

ⓐ $$\{\begin{array}{c}y=4x+9\hfill \\ 5x-2y=\mathrm{-21}\hfill \end{array}$$ ⓑ $$\{\begin{array}{c}9x-4y=24\hfill \\ 3x+5y=\mathrm{-14}\hfill \end{array}$$

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.

Norris can row 3 miles upstream against the current in 1 hour, the same amount of time it takes him to row 5 miles downstream, with the current. Solve the system. $$\{\begin{array}{c}r-c=3\hfill \\ r+c=5\hfill \end{array}$$

ⓐ for $r$, his rowing speed in still water.

ⓑ Then solve for $c$, the speed of the river current.

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
$$\{\begin{array}{c}x+y=10\hfill \\ 5x+8y=56\hfill \end{array}$$

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

The sum of twice a number and nine is 31. Find the number.
If you missed this problem, review Example 3.4.

Twins Jon and Ron together earned $96,000 last year. Ron earned $8,000 more than three times what Jon earned. How much did each of the twins earn?
If you missed this problem, review Example 3.11.

Alessio rides his bike $3\frac{1}{2}$ hours at a rate of 10 miles per hour. How far did he ride?
If you missed this problem, review Example 2.58.

Translate to a system of equations:

The sum of two numbers is negative twenty-three. One number is 7 less than the other. Find the numbers.

Translate to a system of equations:

The sum of two numbers is negative eighteen. One number is 40 more than the other. Find the numbers.

Translate to a system of equations:

A couple has a total household income of $84,000. The husband earns $18,000 less than twice what the wife earns. How much does the wife earn?

Translate to a system of equations:

A senior employee makes $5 less than twice what a new employee makes per hour. Together they make $43 per hour. How much does each employee make per hour?

Translate to a system of equations and then solve:

Ali is 12 years older than his youngest sister, Jameela. The sum of their ages is 40. Find their ages.

Translate to a system of equations and then solve:

Jake’s dad is 6 more than 3 times Jake’s age. The sum of their ages is 42. Find their ages.

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?

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?

Translate to a system of equations and then solve:

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

Translate to a system of equations and then solve:

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

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.

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.

Translate to a system of equations and then solve:

Mario wants to put a rectangular 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.

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.

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?

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 (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?

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.

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.

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 1035 miles in 5 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Translate to a system of equations and then solve: A commercial jet can fly 1728 miles in 4 hours with a tailwind but only 1536 miles in 4 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

Translate to a System of Equations

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

The sum of two numbers is fifteen. One number is three less than the other. Find the numbers.

Translate to a System of Equations

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

The sum of two numbers is negative thirty. One number is five times the other. Find the numbers.

Translate to a System of Equations

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

Twice a number plus three times a second number is twenty-two. Three times the first number plus four times the second is thirty-one. Find the numbers.

Translate to a System of Equations

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

Three times a number plus three times a second number is fifteen. Four times the first plus twice the second number is fourteen. Find the numbers.

Translate to a System of Equations

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

A married couple together earn $75,000. The husband earns $15,000 more than five times what his wife earns. What does the wife earn?

Translate to a System of Equations

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

Daniela invested a total of $50,000, some in a certificate of deposit (CD) and the remainder in bonds. The amount invested in bonds was $5000 more than twice the amount she put into the CD. How much did she invest in each account?

Translate to a System of Equations

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

In her last two years in college, Marlene received $42,000 in loans. The first year she received a loan that was $6,000 less than three times the amount of the second year’s loan. What was the amount of her loan for each year?

Solve Direct Translation Applications

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

Alyssa is twelve years older than her sister, Bethany. The sum of their ages is forty-four. Find their ages.

Solve Direct Translation Applications

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

The age of Noelle’s dad is six less than three times Noelle’s age. The sum of their ages is seventy-four. Find their ages.

Solve Direct Translation Applications

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

Two containers of gasoline hold a total of fifty gallons. The big container can hold ten gallons less than twice the small container. How many gallons does each container hold?

Solve Direct Translation Applications

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

Shelly spent 10 minutes jogging and 20 minutes cycling and burned 300 calories. The next day, Shelly swapped times, doing 20 minutes of jogging and 10 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?

Solve Direct Translation Applications

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

Troy and Lisa were shopping for school supplies. Each purchased different quantities of the same notebook and thumb drive. Troy bought four notebooks and five thumb drives for $116. Lisa bought two notebooks and three thumb dives for $68. Find the cost of each notebook and each thumb drive.

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

The difference of two complementary angles is 30 degrees. Find the measures of the angles.

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

The difference of two supplementary angles is 70 degrees. Find the measures of the angles.

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

The difference of two supplementary angles is 8 degrees. Find the measures of the angles.

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

The difference of two complementary angles is 55 degrees. Find the measures of the angles.

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 four more than three times the measure of the smaller angle. Find the measures of both angles.

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 twelve less than twice the measure of the smaller angle. Find the measures of both angles.

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

Wayne is hanging a string of lights 45 feet long around the three sides of his rectangular patio, which is adjacent to his house. The length of his patio, the side along the house, is five feet longer than twice its width. Find the length and width of the patio.

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

A frame around a rectangular family portrait has a perimeter of 60 inches. The length is fifteen less than twice the width. Find the length and width of the frame.