Chapter 2: Solving Linear EquationsContent check — Intermediate Algebra, Chapter 2: Solving Linear Equations

Simplify: $\frac{3}{2}\left(12x+20\right).$
If you missed this problem, review Example 1.51.

Simplify: $5-2\left(n+1\right).$
If you missed this problem, review Example 1.55.

Find the LCD of $\frac{5}{6}$ and $\frac{1}{4}.$
If you missed this problem, review Example 1.28.

Determine whether the values are solutions to the equation: $9y+2=6y+3.$

ⓐ $y=\frac{4}{3}$ ⓑ $y=\frac{1}{3}$

Determine whether the values are solutions to the equation: $4x-2=2x+1.$

ⓐ $x=\frac{3}{2}$ ⓑ $x=-\frac{1}{2}$

Solve: $2(m-4)+$ $3=$ $\mathrm{-1}.$

Solve: $5(a-3)+$ $5=$ $\mathrm{-10}.$

Solve: $\frac{1}{3}(6u+3)=$ $7-$ $u.$

Solve: $\frac{2}{3}(9x-12)=$ $8+$ $2x.$

Solve: $6(p-3)-$ $7=$ $5(4p+3)-$ $12.$

Solve: $8(q+1)-$ $5=$ $3(2q-4)-$ $1.$

Solve: $6[4-2(7y-1\left)\right]=$ $8(13-8y).$

Solve: $12[1-5(4z-1\left)\right]=$ $3(24+11z).$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: $4+$ $9(3x-7)=$ $\mathrm{-42}x-$ $13+$ $23(3x-2).$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: $8(1-3x)+$ $15(2x+7)=$ $2(x+50)+$ $4(x+3)+$ $1.$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: $11(q+3)-$ $5=$ $19.$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: $\mathrm{6+14}(k-8)=$ $95.$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: $12c+$ $5(5+3c)=$ $3(9c-4).$

Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: $4(7d+18)=$ $13(3d-2)-$ $11d.$

Solve: $\frac{1}{4}x+\frac{1}{2}=\frac{5}{8}.$

Solve: $\frac{1}{8}x+\frac{1}{2}=\frac{1}{4}.$

Solve: $7=$ $\frac{1}{2}x+$ $\frac{3}{4}x-$ $\frac{2}{3}x.$

Solve: $\mathrm{-1}=$ $\frac{1}{2}u+$ $\frac{1}{4}u-$ $\frac{2}{3}u.$

Solve: $\frac{1}{5}(n+3)=$ $\frac{1}{4}(n+2).$

Solve: $\frac{1}{2}(m-3)=$ $\frac{1}{4}(m-7).$

Solve: $\frac{3r+5}{6}+$ $1=$ $\frac{4r+3}{3}.$

Solve: $\frac{2s+3}{2}+$ $1=$ $\frac{3s+2}{4}.$

Solve: $0.25n+$ $0.05(n+5)=$ $2.95.$

Solve: $0.10d+$ $0.05(d-5)=$ $2.15.$

Solve Equations Using the General Strategy

In the following exercises, determine whether the given values are solutions to the equation.

$6y+10=12y$

ⓐ $y=\frac{5}{3}$ ⓑ $y=-\frac{1}{2}$

Solve Equations Using the General Strategy

In the following exercises, determine whether the given values are solutions to the equation.

$8u-1=6u$

ⓐ $u=-\frac{1}{2}$ ⓑ $u=\frac{1}{2}$

In the following exercises, solve each linear equation.

$15\left(y-9\right)=\mathrm{-60}$

In the following exercises, solve each linear equation.

$\text{−}\left(w-12\right)=30$

In the following exercises, solve each linear equation.

$51+$ $5(4-q)=$ $56$

In the following exercises, solve each linear equation.

$3(10-2x)+$ $54=$ $0$

In the following exercises, solve each linear equation.

$\frac{2}{3}\left(9c-3\right)=22$

In the following exercises, solve each linear equation.

$\frac{1}{5}(15c+10)=$ $c+$ $7$

In the following exercises, solve each linear equation.

$3(4n-1)-$ $2=$ $8n+$ $3$

In the following exercises, solve each linear equation.

$12+$ $2(5-3y)=$ $\mathrm{-9}(y-1)-$ $2$

In the following exercises, solve each linear equation.

$5+$ $6(3s-5)=$ $\mathrm{-3}+$ $2(8s-1)$

In the following exercises, solve each linear equation.

$4(p-4)-$ $(p+7)=$ $5(p-3)$

In the following exercises, solve each linear equation.

$4[5-8(4c-3\left)\right]=$ $12(1-13c)-$ $8$

In the following exercises, solve each linear equation.

$3[\mathrm{-9}+8(4h-3\left)\right]=$ $2(5-12h)-$ $19$

In the following exercises, solve each linear equation.

$5\left[2\right(m+4)+8(m-7\left)\right]=$ $2\left[3\right(5+m)-(21-3m\left)\right]$

Classify Equations

In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$23z+$ $19=$ $3(5z-9)+$ $8z+$ $46$

Classify Equations

In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$18(5j-1)+$ $29=$ $47$

Classify Equations

In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$22(3m-4)=$ $8(2m+9)$

Classify Equations

In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$7v+$ $42=$ $11(3v+8)-$ $2(13v-1)$

Classify Equations

In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$45(3y-2)=$ $9(15y-6)$

Classify Equations

In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$9(14d+9)+$ $4d=$ $13(10d+6)+$ $3$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{1}{4}x-\frac{1}{2}=-\frac{3}{4}$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{5}{6}y-\frac{2}{3}=-\frac{3}{2}$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{1}{2}a+\frac{3}{8}=\frac{3}{4}$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$2=$ $\frac{1}{3}x-$ $\frac{1}{2}x+$ $\frac{2}{3}x$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{1}{3}w+\frac{5}{4}=w-\frac{1}{4}$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{1}{3}b+$ $\frac{1}{5}=$ $\frac{2}{5}b-$ $\frac{3}{5}$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{1}{4}(p-7)=$ $\frac{1}{3}(p+5)$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{1}{2}\left(x+4\right)=\frac{3}{4}$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{4n+8}{4}=\frac{n}{3}$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{3x+4}{2}+$ $1=$ $\frac{5x+10}{8}$

Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{7u-1}{4}-$ $1=$ $\frac{4u+8}{5}$

In the following exercises, solve each equation with decimal coefficients.

$0.4x+$ $0.6=$ $0.5x-$ $1.2$

In the following exercises, solve each equation with decimal coefficients.

$0.9x-$ $1.25=$ $0.75x+$ $1.75$

In the following exercises, solve each equation with decimal coefficients.

$0.05n+$ $0.10(n+8)=$ $2.15$

In the following exercises, solve each equation with decimal coefficients.

$0.10d+$ $0.25(d+5)=$ $4.05$

In the following exercises, solve each equation with decimal coefficients.

Fencing Micah has 74 feet of fencing to make a dog run in his yard. He wants the length to be 2.5 feet more than the width. Find the length, L, by solving the equation $2L+$ $2(L-2.5)=$ $74.$

In the following exercises, solve each equation with decimal coefficients.

Using your own words, list the steps in the general strategy for solving linear equations.

In the following exercises, solve each equation with decimal coefficients.

What is the first step you take when solving the equation $3-$ $7(y-4)=$ $38?$ Why is this your first step?

In the following exercises, solve each equation with decimal coefficients.

If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?

Translate “six less than twice x” into an algebraic expression.
If you missed this problem, review Example 1.8.

Convert 4.5% to a decimal.
If you missed this problem, review Example 1.40.

Convert 0.6 to a percent.
If you missed this problem, review Example 1.40.

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was three more than twice the number of notebooks. He bought seven textbooks. How many notebooks did he buy?

Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?

The sum of four times a number and two is fourteen. Find the number.

The sum of three times a number and seven is twenty-five. Find the number.

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

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

Find three consecutive integers whose sum is $\mathrm{-96}.$

Find three consecutive integers whose sum is $\mathrm{-36}.$

Find three consecutive even integers whose sum is 102.

Find three consecutive even integers whose sum is $\mathrm{-24}.$

According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,400. This was $1,600 less than six times the cost in 1975. What was the average cost of a car in 1975?

US Census data shows that the median price of new home in the U.S. in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

Translate and solve: ⓐ What number is 45% of 80? ⓑ 7.5% of what amount is $1.95? ⓒ 110 is what percent of 88?

Translate and solve: ⓐ What number is 55% of 60? ⓑ 8.5% of what amount is $3.06? ⓒ 126 is what percent of 72?

One serving of wheat square cereal has 7 grams of fiber, which is 28% of the recommended daily amount. What is the total recommended daily amount of fiber?

One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium?

Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was 480 calories, and had 240 calories of fat. What percent of the total calories in each brownie comes from fat? Round the answer to the nearest whole percent.

The mix Ricardo plans to use to make brownies says that each brownie will be 190 calories, and 76 calories are from fat. What percent of the total calories are from fat? Round the answer to the nearest whole percent.

Find the percent change. (Round to the nearest tenth of a percent.) In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents.

Find the percent change. (Round to the nearest tenth of a percent.) In 1995, the standard bus fare in Chicago was $1.50. In 2008, the standard bus fare was 2.25.

Find ⓐ the amount of mark-up and ⓑ the list price: Jim’s music store bought a guitar at original cost $1,200. Jim marked the price up 50%.

Find ⓐ the amount of mark-up and ⓑ the list price: The Auto Resale Store bought Pablo’s Toyota for $8,500. They marked the price up 35%.

Nathaly deposited $12,500 in her bank account where it will earn 4% simple interest. How much interest will Nathaly earn in five years?

Susana invested a principal of $36,000 in her bank account that earned simple interest at an interest rate of $6.5\%.$ How much interest did she earn in three years?

Jim lent his sister $5,000 to help her buy a house. In three years, she paid him the $5,000, plus $900 interest. What was the rate of simple interest?

Loren lent his brother $3,000 to help him buy a car. In four years, his brother paid him back the $3,000 plus $660 in interest. What was the rate of simple interest?

Eduardo noticed that his new car loan papers stated that with a 7.5% simple interest rate, he would pay $6,596.25 in interest over five years. How much did he borrow to pay for his car?

In five years, Gloria’s bank account earned $2,400 interest at 5% simple interest. How much had she deposited in the account?

Use a Problem Solving Strategy for Word Problems

List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?

Solve Number Word Problems

In the following exercises, solve each number word problem.

The difference of a number and 12 is three. Find the number.

Solve Number Word Problems

In the following exercises, solve each number word problem.

The sum of three times a number and eight is 23. Find the number.

Solve Number Word Problems

In the following exercises, solve each number word problem.

The difference of twice a number and seven is 17. Find the number.

Solve Number Word Problems

In the following exercises, solve each number word problem.

Three times the sum of a number and nine is 12. Find the number.

Solve Number Word Problems

In the following exercises, solve each number word problem.

One number is six more than the other. Their sum is 42. Find the numbers.

Solve Number Word Problems

In the following exercises, solve each number word problem.

The sum of two numbers is 20. One number is four less than the other. Find the numbers.

Solve Number Word Problems

In the following exercises, solve each number word problem.

One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

Solve Number Word Problems

In the following exercises, solve each number word problem.

The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

Solve Number Word Problems

In the following exercises, solve each number word problem.

The sum of two consecutive integers is 77. Find the integers.

Solve Number Word Problems

In the following exercises, solve each number word problem.

The sum of three consecutive integers is 78. Find the integers.

Solve Number Word Problems

In the following exercises, solve each number word problem.

Find three consecutive integers whose sum is $\mathrm{-36}.$

Solve Number Word Problems

In the following exercises, solve each number word problem.

Find three consecutive even integers whose sum is 258.

Solve Number Word Problems

In the following exercises, solve each number word problem.

Find three consecutive odd integers whose sum is $\mathrm{-213}.$

Solve Number Word Problems

In the following exercises, solve each number word problem.

Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

Solve Number Word Problems

In the following exercises, solve each number word problem.

Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

Solve Percent Applications

In the following exercises, translate and solve.

ⓐ What number is 45% of 120? ⓑ 81 is 75% of what number? ⓐ What percent of 260 is 78?

Solve Percent Applications

In the following exercises, translate and solve.

ⓐ 250% of 65 is what number? ⓑ 8.2% of what amount is $2.87? ⓐ 30 is what percent of 20?

In the following exercises, solve.

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16% of the total bill as a tip. How much should the tip be?

In the following exercises, solve.

One serving of oatmeal has 8 grams of fiber, which is 33% of the recommended daily amount. What is the total recommended daily amount of fiber?

In the following exercises, solve.

A bacon cheeseburger at a popular fast food restaurant contains 2070 milligrams (mg) of sodium, which is 86% of the recommended daily amount. What is the total recommended daily amount of sodium?

In the following exercises, solve.

The nutrition fact sheet at a fast food restaurant says the fish sandwich has 380 calories, and 171 calories are from fat. What percent of the total calories is from fat?

In the following exercises, solve.

Emma gets paid $3,000 per month. She pays $750 a month for rent. What percent of her monthly pay goes to rent?

In the following exercises, solve.

Tamanika received a raise in her hourly pay, from $15.50 to $17.36. Find the percent change.

In the following exercises, solve.

Annual student fees at the University of California rose from about $4,000 in 2000 to about $12,000 in 2010. Find the percent change.

In the following exercises, solve.

A grocery store reduced the price of a loaf of bread from $2.80 to $2.73. Find the percent change.

In the following exercises, solve.

Hernando’s salary was $49,500 last year. This year his salary was cut to $44,055. Find the percent change.

In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.

Janelle bought a beach chair on sale at 60% off. The original price was $44.95.

In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate (Round to the nearest tenth of a percent if needed.)

Larry and Donna bought a sofa at the sale price of $1,344. The original price of the sofa was $1,920.

In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

Daria bought a bracelet at original cost $16 to sell in her handicraft store. She marked the price up 45%. What was the list price of the bracelet?

In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

Tom paid $0.60 a pound for tomatoes to sell at his produce store. He added a 33% mark-up. What price did he charge his customers for the tomatoes?

Solve Simple Interest Applications

In the following exercises, solve.

Casey deposited $1,450 in a bank account that earned simple interest at an interest rate of 4%. How much interest was earned in two years?

Solve Simple Interest Applications

In the following exercises, solve.

Robin deposited $31,000 in a bank account that earned simple interest at an interest rate of 5.2%. How much interest was earned in three years?

Solve Simple Interest Applications

In the following exercises, solve.

Hilaria borrowed $8,000 from her grandfather to pay for college. Five years later, she paid him back the $8,000, plus $1,200 interest. What was the rate of simple interest?

Solve Simple Interest Applications

In the following exercises, solve.

Lebron lent his daughter $20,000 to help her buy a condominium. When she sold the condominium four years later, she paid him the $20,000, plus $3,000 interest. What was the rate of simple interest?

Solve Simple Interest Applications

In the following exercises, solve.

In 10 years, a bank account that paid 5.25% simple interest earned $18,375 interest. What was the principal of the account?

Solve Simple Interest Applications

In the following exercises, solve.

Joshua’s computer loan statement said he would pay $1,244.34 in simple interest for a three-year loan at 12.4%. How much did Joshua borrow to buy the computer?

Solve Simple Interest Applications

In the following exercises, solve.

Tipping At the campus coffee cart, a medium coffee costs $1.65. MaryAnne brings $2.00 with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave?

Solve Simple Interest Applications

In the following exercises, solve.

What has been your past experience solving word problems? Where do you see yourself moving forward?

Solve Simple Interest Applications

In the following exercises, solve.

After returning from vacation, Alex said he should have packed 50% fewer shorts and 200% more shirts. Explain what Alex meant.

Evaluate $2(x+3)$ when $x=5.$
If you missed this problem, review Example 1.6.

The length of a rectangle is three less than the width. Let w represent the width. Write an expression for the length of the rectangle.
If you missed this problem, review Example 1.10.

Evaluate $\frac{1}{2}bh$ when $b=14$ and $h=9.$
If you missed this problem, review Example 1.33.

Use the formula $A=\frac{1}{2}bh$ to solve for b.

Use the formula $A=\frac{1}{2}bh$ to solve for h.

Solve the formula $F=\frac{9}{5}C+32$ for C.

Solve the formula $A=\frac{1}{2}h\left(b+B\right)$ for b.

Solve the formula $A=P+Prt$ for t.

Solve the formula $A=P+Prt$ for r.

Solve the formula $4x+7y=9$ for y.

Solve the formula $5x+8y=1$ for y.

The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window’s height?

A triangular tent door has an area of 15 square feet. The height is five feet. What is the length of the base?

The measure of one angle of a right triangle is 50 more than the measure of the smallest angle. Find the measures of all three angles.

The measure of one angle of a right triangle is 30 more than the measure of the smallest angle. Find the measures of all three angles.

Use the Pythagorean Theorem to find the length of the leg in the figure.

Use the Pythagorean Theorem to find the length of the leg in the figure.

The length of a rectangle is seven more than twice the width. The perimeter is 110 inches. Find the length and width.

The width of a rectangle is eight yards less than twice the length. The perimeter is 86 yards. Find the length and width.

One side of a triangle is seven inches more than the first side. The third side is four inches less than three times the first. The perimeter is 28 inches. Find the length of the three sides of the triangle.

One side of a triangle is three feet less than the first side. The third side is five feet less than twice the first. The perimeter is 20 feet. Find the length of the three sides of the triangle.

The perimeter of a rectangular swimming pool is 200 feet. The length is 40 feet more than the width. Find the length and width.

The length of a rectangular garden is 30 yards more than the width. The perimeter is 300 yards. Find the length and width.

John puts the base of a 13-foot ladder five feet from the wall of his house as shown in the figure. How far up the wall does the ladder reach?

Randy wants to attach a 17-foot string of lights to the top of the 15 foot mast of his sailboat, as shown in the figure. How far from the base of the mast should he attach the end of the light string?

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

Solve the formula $C=\pi d$ for d.

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

Solve the formula $V=LWH$ for L.

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

Solve the formula $A=\frac{1}{2}bh$ for b.

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

Solve the formula
$A=\frac{1}{2}{d}_1{d}_2$ for $d_1.$

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

Solve the formula
$A=$ $\frac{1}{2}h(b_1+b_2)$ for $b_1.$

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

Solve the formula
$h=54t+\frac{1}{2}a{t}^2$ for a.

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

Solve $180=a+b+c$ for a.

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

Solve the formula
$A=\frac{1}{2}pl+B$ for p.

Solve a Formula for a Specific Variable

In the following exercises, solve the given formula for the specified variable.

Solve the formula
$P=2L+2W$ for L.

In the following exercises, solve for the formula for y.

Solve the formula
$8x+y=15$ for y.

In the following exercises, solve for the formula for y.

Solve the formula
$\mathrm{-4}x+y=\mathrm{-6}$ for y.

In the following exercises, solve for the formula for y.

Solve the formula
$x-y=\mathrm{-4}$ for y.

In the following exercises, solve for the formula for y.

Solve the formula
$4x+3y=7$ for y.

In the following exercises, solve for the formula for y.

Solve the formula
$2x+3y=12$ for y.

In the following exercises, solve for the formula for y.

Solve the formula
$3x-2y=18$ for y.

Use Formulas to Solve Geometry Applications

In the following exercises, solve using a geometry formula.

A triangular flag has area 0.75 square feet and height 1.5 foot. What is its base?

Use Formulas to Solve Geometry Applications

In the following exercises, solve using a geometry formula.

What is the base of a triangle with area 207 square inches and height 18 inches?

Use Formulas to Solve Geometry Applications

In the following exercises, solve using a geometry formula.

The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.

Use Formulas to Solve Geometry Applications

In the following exercises, solve using a geometry formula.

The angles in a triangle are such that one angle is twice the smallest angle, while the third angle is three times as large as the smallest angle. Find the measures of all three angles.

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth if necessary.

In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth if necessary.

In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth if necessary.

In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth if necessary.

In the following exercises, solve using a geometry formula.

The width of a rectangle is seven meters less than the length. The perimeter is 58 meters. Find the length and width.

In the following exercises, solve using a geometry formula.

The width of the rectangle is 0.7 meters less than the length. The perimeter of a rectangle is 52.6 meters. Find the dimensions of the rectangle.

In the following exercises, solve using a geometry formula.

The perimeter of a rectangle of 150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.

In the following exercises, solve using a geometry formula.

The length of the rectangle is three meters less than twice the width. The perimeter of a rectangle is 36 meters. Find the dimensions of the rectangle.

In the following exercises, solve using a geometry formula.

The perimeter of a triangle is 39 feet. One side of the triangle is one foot longer than the second side. The third side is two feet longer than the second side. Find the length of each side.

In the following exercises, solve using a geometry formula.

One side of a triangle is twice the smallest side. The third side is five feet more than the shortest side. The perimeter is 17 feet. Find the lengths of all three sides.

In the following exercises, solve using a geometry formula.

The perimeter of a rectangular field is 560 yards. The length is 40 yards more than the width. Find the length and width of the field.

In the following exercises, solve using a geometry formula.

A rectangular parking lot has perimeter 250 feet. The length is five feet more than twice the width. Find the length and width of the parking lot.

In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.

A 13-foot string of lights will be attached to the top of a 12-foot pole for a holiday display as shown. How far from the base of the pole should the end of the string of lights be anchored?

In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.

Chi is planning to put a diagonal path of paving stones through her flower garden as shown. The flower garden is a square with side 10 feet. What will the length of the path be?

In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.

Converting temperature While on a tour in Greece, Tatyana saw that the temperature was 40° Celsius. Solve for F in the formula $C=\frac{5}{9}\left(F-32\right)$ to find the Fahrenheit temperature.

In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.

Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are six feet, eight feet and 10 feet. How many feet of fencing will she need to enclose her flowerbed?

In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.

If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.

In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.

Look at the two figures below.

ⓐ Which figure looks like it has the larger area? Which looks like it has the larger perimeter?
ⓑ Now calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter?
ⓒ Were the results of part (b) the same as your answers in part (a)? Is that surprising to you?

Simplify: $0.25x+$ $0.10(x+4)\mathrm{.}$
If you missed this problem, review Example 1.52.

The number of adult tickets is three more than twice the number of children tickets. Let c represent the number of children tickets. Write an expression for the number of adult tickets.
If you missed this problem, review Example 1.11.

Convert 4.2% to a decimal.
If you missed this problem, review Example 1.40.

Jesse has $6.55 worth of quarters and nickels in his pocket. The number of nickels is five more than two times the number of quarters. How many nickels and how many quarters does Jesse have?

Elane has $7.00 total in dimes and nickels in her coin jar. The number of dimes that Elane has is seven less than three times the number of nickels. How many of each coin does Elane have?

Eric paid $19.88 for stamps. The number of 49-cent stamps was eight more than twice the number of 35-cent stamps. How many 49-cent stamps and how many 35-cent stamps did Eric buy?

Kailee paid $14.74 for stamps. The number of 49-cent stamps was four less than three times the number of 20-cent stamps. How many 49-cent stamps and how many 20-cent stamps did Kailee buy?

During her shift at the museum ticket booth, Leah sold 115 tickets for a total of $1,163. Adult tickets cost $12 and student tickets cost $5. How many adult tickets and how many student tickets did Leah sell?

Galen sold 810 tickets for his church’s carnival for a total revenue of $2,820. Children’s tickets cost $3 each and adult tickets cost $5 each. How many children’s tickets and how many adult tickets did he sell?

Orlando is mixing nuts and cereal squares to make a party mix. Nuts sell for $7 a pound and cereal squares sell for $4 a pound. Orlando wants to make 30 pounds of party mix at a cost of $6.50 a pound, how many pounds of nuts and how many pounds of cereal squares should he use?

Becca wants to mix fruit juice and soda to make a punch. She can buy fruit juice for $3 a gallon and soda for $4 a gallon. If she wants to make 28 gallons of punch at a cost of $3.25 a gallon, how many gallons of fruit juice and how many gallons of soda should she buy?

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.

Jeromy can drive from his house in Cleveland to his college in Chicago in 4.5 hours. It takes his mother six hours to make the same drive. Jeromy drives 20 miles per hour faster than his mother. Find Jeromy’s speed and his mother’s speed.

Christopher and his parents live 115 miles apart. They met at a restaurant between their homes to celebrate his mother’s birthday. Christopher drove one and a half hours while his parents drove one hour to get to the restaurant. Christopher’s average speed was ten miles per hour faster than his parents’ average speed. What were the average speeds of Christopher and of his parents as they drove to the restaurant?

Ashley goes to college in Minneapolis, 234 miles from her home in Sioux Falls. She wants her parents to bring her more winter clothes, so they decide to meet at a restaurant on the road between Minneapolis and Sioux Falls. Ashley and her parents both drove two hours to the restaurant. Ashley’s average speed was seven miles per hour faster than her parents’ average speed. Find Ashley’s and her parents’ average speed.

Pierre and Monique leave their home in Portland at the same time. Pierre drives north on the turnpike at a speed of 75 miles per hour while Monique drives south at a speed of 68 miles per hour. How long will it take them to be 429 miles apart?

Thanh and Nhat leave their office in Sacramento at the same time. Thanh drives north on I-5 at a speed of 72 miles per hour. Nhat drives south on I-5 at a speed of 76 miles per hour. How long will it take them to be 330 miles apart?

Suzy takes 50 minutes to hike uphill from the parking lot to the lookout tower. It takes her 30 minutes to hike back down to the parking lot. Her speed going downhill is 1.2 miles per hour faster than her speed going uphill. Find Suzy’s uphill and downhill speeds.

Llewyn takes 45 minutes to drive his boat upstream from the dock to his favorite fishing spot. It takes him 30 minutes to drive the boat back downstream to the dock. The boat’s speed going downstream is four miles per hour faster than its speed going upstream. Find the boat’s upstream and downstream speeds.

Hamilton loves to travel to Las Vegas, 255 miles from his home in Orange County. On his last trip, he left his house at 2:00 p.m. The first part of his trip was on congested city freeways. At 4:00 pm, the traffic cleared and he was able to drive through the desert at a speed 1.75 times as fast as when he drove in the congested area. He arrived in Las Vegas at 6:30 p.m. How fast was he driving during each part of his trip?

Phuong left home on his bicycle at 10:00. He rode on the flat street until 11:15, then rode uphill until 11:45. He rode a total of 31 miles. His speed riding uphill was 0.6 times his speed on the flat street. Find his speed biking uphill and on the flat street.

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

Michaela has $2.05 in dimes and nickels in her change purse. She has seven more dimes than nickels. How many coins of each type does she have?

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

In a cash drawer there is $125 in $5 and $10 bills. The number of $10 bills is twice the number of $5 bills. How many of each type of bill is in the drawer?

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

Chi has $11.30 in dimes and quarters. The number of dimes is three more than three times the number of quarters. How many of each are there?

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

Mukul has $3.75 in quarters, dimes and nickels in his pocket. He has five more dimes than quarters and nine more nickels than quarters. How many of each coin are in his pocket?

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

The first day of a water polo tournament the total value of tickets sold was $17,610. One-day passes sold for $20 and tournament passes sold for $30. The number of tournament passes sold was 37 more than the number of day passes sold. How many day passes and how many tournament passes were sold?

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

Julie went to the post office and bought both $0.41 stamps and $0.26 postcards. She spent $51.40. The number of stamps was 20 more than twice the number of postcards. How many of each did she buy?

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

Hilda has $210 worth of $10 and $12 stock shares. The number of $10 shares is five more than twice the number of $12 shares. How many of each type of share does she have?

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

The ice rink sold 95 tickets for the afternoon skating session, for a total of $828. General admission tickets cost $10 each and youth tickets cost $8 each. How many general admission tickets and how many youth tickets were sold?

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

The box office sold 360 tickets to a concert at the college. The total receipts were $4,170. General admission tickets cost $15 and student tickets cost $10. How many of each kind of ticket was sold?

Solve Mixture Word Problems

In the following exercises, solve each mixture word problem.

Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. Macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. How many pounds of macadamia nuts and how many pounds of almonds should Macario use for the mixture to cost $6.50 per pound to make?

Solve Mixture Word Problems

In the following exercises, solve each mixture word problem.

Riley is planning to plant a lawn in his yard. He will need nine pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs $3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound?

Solve Mixture Word Problems

In the following exercises, solve each mixture word problem.

Vern sold his 1964 Ford Mustang for $55,000 and wants to invest the money to earn him 5.8% interest per year. He will put some of the money into Fund A that earns 3% per year and the rest in Fund B that earns 10% per year. How much should he invest into each fund if he wants to earn 5.8% interest per year on the total amount?

Solve Mixture Word Problems

In the following exercises, solve each mixture word problem.

Liam borrowed a total of $35,000 to pay for college. He pays his parents 3% interest on the $8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total $35,000? (Round your answer to the nearest tenth of a percent.)

Solve Uniform Motion Applications

In the following exercises, solve.

Kathy and Cheryl are walking in a fundraiser. Kathy completes the course in 4.8 hours and Cheryl completes the course in eight hours. Kathy walks two miles per hour faster than Cheryl. Find Kathy’s speed and Cheryl’s speed.

Solve Uniform Motion Applications

In the following exercises, solve.

A commercial jet and a private airplane fly from Denver to Phoenix. It takes the commercial jet 1.6 hours for the flight, and it takes the private airplane 2.6 hours. The speed of the commercial jet is 210 miles per hour faster than the speed of the private airplane. Find the speed of both airplanes to the nearest 10 mph.

Solve Uniform Motion Applications

In the following exercises, solve.

Charlie and Violet met for lunch at a restaurant between Memphis and New Orleans. Charlie had left Memphis and drove 4.8 hours towards New Orleans. Violet had left New Orleans and drove two hours towards Memphis, at a speed 10 miles per hour faster than Charlie’s speed. The distance between Memphis and New Orleans is 394 miles. Find the speed of the two drivers.

Solve Uniform Motion Applications

In the following exercises, solve.

Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden two hours, they meet on the path. Ethan’s speed is six miles per hour faster than Leo’s speed. Find the speed of the two bikers.

Solve Uniform Motion Applications

In the following exercises, solve.

DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players.

Solve Uniform Motion Applications

In the following exercises, solve.

Matt and Chris leave their uncle’s house in Phoenix at the same time. Matt drives west on I-60 at a speed of 76 miles per hour. Chris drives east on I-60 at a speed of 82 miles per hour. How many hours will it take them to be 632 miles apart?

Solve Uniform Motion Applications

In the following exercises, solve.

Two boats leave the same dock in Cairo at the same time. One heads north on the Mississippi River while the other heads south. The northbound boat travels four miles per hour. The southbound boat goes eight miles per hour. How long will it take them to be 54 miles apart?

Solve Uniform Motion Applications

In the following exercises, solve.

Julian rides his bike uphill for 45 minutes, then turns around and rides back downhill. It takes him 15 minutes to get back to where he started. His uphill speed is 3.2 miles per hour slower than his downhill speed. Find Julian’s uphill and downhill speed.

Solve Uniform Motion Applications

In the following exercises, solve.

It takes Darline 20 minutes to drive to work in light traffic. To come home, when there is heavy traffic, it takes her 36 minutes. Her speed in light traffic is 24 miles per hour faster than her speed in heavy traffic. Find her speed in light traffic and in heavy traffic.

Solve Uniform Motion Applications

In the following exercises, solve.

Aaron left at 9:15 to drive to his mountain cabin 108 miles away. He drove on the freeway until 10:45 and then drove on a mountain road. He arrived at 11:05. His speed on the freeway was three times his speed on the mountain road. Find Aaron’s speed on the freeway and on the mountain road.

Solve Uniform Motion Applications

In the following exercises, solve.

Lizette is training for a marathon. At 7:00 she left her house and ran until 8:15 then she walked until 11:15. She covered a total distance of 19 miles. Her running speed was five miles per hour faster than her walking speed. Find her running and walking speeds.

Solve Uniform Motion Applications

In the following exercises, solve.

Sarah wants to arrive at her friend’s wedding at 3:00. The distance from Sarah’s house to the wedding is 95 miles. Based on usual traffic patterns, Sarah predicts she can drive the first 15 miles at 60 miles per hour, the next 10 miles at 30 miles per hour, and the remainder of the drive at 70 miles per hour.

ⓐ How long will it take Sarah to drive the first 15 miles?

ⓑ How long will it take Sarah to drive the next 10 miles?

ⓒ How long will it take Sarah to drive the rest of the trip?

ⓓ What time should Sarah leave her house?

Solve Uniform Motion Applications

In the following exercises, solve.

Do you find it helpful to use a table when solving coin problems? Why or why not?

Solve Uniform Motion Applications

In the following exercises, solve.

When solving a uniform motion problem, how does drawing a diagram of the situation help you?

Translate from algebra to English: $15>x.$
If you missed this problem, review Example 1.3.

Translate to an algebraic expression: 15 is less than x.
If you missed this problem, review Example 1.8.

Graph each inequality on the number line and write in interval notation: ⓐ $x>2$ ⓑ $x\le \mathrm{-1.5}$ ⓒ $x\ge \frac{3}{4}.$

Graph each inequality on the number line and write in interval notation: ⓐ $x\le \mathrm{-4}$ ⓑ $x\ge 0.5$ ⓒ $x<-\frac{2}{3}.$

Graph each inequality on the number line and write in interval notation:

ⓐ $\mathrm{-2}<$ $x<$ $1$ ⓑ $\mathrm{-5}\le x<\mathrm{-4}$ ⓒ $1\le x\le 4.25$

Graph each inequality on the number line and write in interval notation:

ⓐ $\mathrm{-6}<$ $x<$ $2$ ⓑ $\mathrm{-3}\le x<\mathrm{-1}$ ⓒ $2.5\le x\le 6$

Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

ⓐ $p-\frac{3}{4}\ge \frac{1}{6}$ ⓑ $9c>72$ ⓒ $24\le \frac{3}{8}m$

Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

ⓐ $r-\frac{1}{3}\le \frac{7}{12}$ ⓑ $12d\le \text{}60$ ⓒ $\mathrm{-24}<\frac{4}{3}n$

Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

ⓐ $\mathrm{-8}q<32$ ⓑ $\frac{k}{\mathrm{-12}}\le 15.$

Solve each inequality, graph the solution on the number line, and write the solution in interval notation:

ⓐ $\mathrm{-7}r\le \text{}-70$ ⓑ $\frac{u}{\mathrm{-4}}\ge \mathrm{-16}.$

Solve the inequality, graph the solution on the number line, and write the solution in interval notation: $3q\ge 7q-23.$

Solve the inequality, graph the solution on the number line, and write the solution in interval notation: $6x<$ $10x+$ $19.$

Solve the inequality $9y+$ $2(y+6)>$ $5y-$ $24$, graph the solution on the number line, and write the solution in interval notation.

Solve the inequality $6u+$ $8(u-1)>$ $10u+$ $32$, graph the solution on the number line, and write the solution in interval notation.

Solve the inequality $4b-$ $3(3-b)>$ $5(b-6)+$ $2b$, graph the solution on the number line, and write the solution in interval notation.

Solve the inequality $9h-$ $7(2-h)<$ $8(h+11)+$ $8h$, graph the solution on the number line, and write the solution in interval notation.

Solve the inequality $\frac{1}{4}x-$ $\frac{1}{12}x>$ $\frac{1}{6}x+$ $\frac{7}{8}$, graph the solution on the number line, and write the solution in interval notation.

Solve the inequality $\frac{2}{5}z-$ $\frac{1}{3}z<$ $\frac{1}{15}z\text{}+$ $\frac{3}{5}$, graph the solution on the number line, and write the solution in interval notation.

Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.

Nineteen less than p is no less than 47.

Translate and solve. Then graph the solution on the number line, and write the solution in interval notation.

Four more than a is at most 15.

Angie has $20 to spend on juice boxes for her son’s preschool picnic. Each pack of juice boxes costs $2.63. What is the maximum number of packs she can buy?