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Chapter 5 Review Exercises

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Chapter 5 Review Exercises

Solve Systems of Equations by Graphing

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.

Exercise 1

{x+3y=92x4y=12

  1. (−3,−2)
  2. (0,−3)
Answer
  1. no
  2. yes
Exercise 2

{x+y=8y=x4

  1. (6,2)
  2. (9,−1)

Solve a System of Linear Equations by Graphing

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

Exercise 3

{3x+y=6x+3y=6

Answer

(3,−3)

This figure shows a graph on an x y-coordinate plane of 3x plus y = 6 and x plus 3y = negative 6.

Exercise 4

{y=x2y=2x2

Exercise 5

{2xy=6y=4

Answer

(5,4)

This figure shows a graph on an x y-coordinate plane of 2x – y = 6 and y = 4.

Exercise 6

{x+4y=1x=3

Exercise 7

{2xy=54x2y=10

Answer

coincident lines

This figure shows a graph on an x y-coordinate plane of 2x – y = 5 and 4x – 2y = 10.

Exercise 8

{x+2y=4y=12x3

Determine the Number of Solutions of a Linear System

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

Exercise 9

{y=25x+22x+5y=10

Answer

infinitely many solutions, consistent system, dependent equations

Exercise 10

{3x+2y=6y=3x+4

Exercise 11

{5x4y=0y=54x5

Answer

no solutions, inconsistent system, independent equations

Exercise 12

{y=34x+16x+8y=8

Solve Applications of Systems of Equations by Graphing

Exercise 13

LaVelle is making a pitcher of caffe mocha. For each ounce of chocolate syrup, she uses five ounces of coffee. How many ounces of chocolate syrup and how many ounces of coffee does she need to make 48 ounces of caffe mocha?

Answer

LaVelle needs 8 ounces of chocolate syrup and 40 ounces of coffee.

Exercise 14

Eli is making a party mix that contains pretzels and chex. For each cup of pretzels, he uses three cups of chex. How many cups of pretzels and how many cups of chex does he need to make 12 cups of party mix?

Solve Systems of Equations by Substitution

Solve a System of Equations by Substitution

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

Exercise 15

{3xy=5y=2x+4

Answer

(−1,2)

Exercise 16

{3x2y=2y=12x+3

Exercise 17

{xy=02x+5y=14

Answer

(−2,−2)

Exercise 18

{y=2x+7y=23x1

Exercise 19

{y=5x5x+y=6

Answer

no solution

Exercise 20

{y=13x+2x+3y=6

Solve Applications of Systems of Equations by Substitution

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

Exercise 21

The sum of two number is 55. One number is 11 less than the other. Find the numbers.

Answer

The numbers are 22 and 33.

Exercise 22

The perimeter of a rectangle is 128. The length is 16 more than the width. Find the length and width.

Exercise 23

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

Answer

The measures are 23 degrees and 67 degrees.

Exercise 24

Gabriela works for an insurance company that pays her a salary of $32,000 plus a commission of $100 for each policy she sells. She is considering changing jobs to a company that would pay a salary of $40,000 plus a commission of $80 for each policy sold. How many policies would Gabriela need to sell to make the total pay the same?

Solve Systems of Equations by Elimination

Solve a System of Equations by Elimination In the following exercises, solve the systems of equations by elimination.

Exercise 25

{x+y=12xy=10

Answer

(1,11)

Exercise 26

{4x+2y=24x3y=9

Exercise 27

{3x8y=20x+3y=1

Answer

(4,−1)

Exercise 28

{3x2y=64x+3y=8

Exercise 29

{9x+4y=25x+3y=5

Answer

(−2,5)

Exercise 30

{x+3y=82x6y=20

Solve Applications of Systems of Equations by Elimination

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

Exercise 31

The sum of two numbers is −90. Their difference is 16. Find the numbers.

Answer

The numbers are −37 and −53.

Exercise 32

Omar stops at a donut shop every day on his way to work. Last week he had 8 donuts and 5 cappuccinos, which gave him a total of 3,000 calories. This week he had 6 donuts and 3 cappuccinos, which was a total of 2,160 calories. How many calories are in one donut? How many calories are in one cappuccino?

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.

Exercise 33

{6x5y=273x+10y=24

Answer

elimination

Exercise 34

{y=3x94x5y=23

Solve Applications with Systems of Equations

Translate to a System of Equations

In the following exercises, translate to a system of equations. Do not solve the system.

Exercise 35

The sum of two numbers is −32. One number is two less than twice the other. Find the numbers.

Answer

{x+y=32x=2y2

Exercise 36

Four times a number plus three times a second number is −9. Twice the first number plus the second number is three. Find the numbers.

Exercise 37

Last month Jim and Debbie earned $7,200. Debbie earned $1,600 more than Jim earned. How much did they each earn?

Answer

{j+d=7200d=j+1600

Exercise 38

Henri has $24,000 invested in stocks and bonds. The amount in stocks is $6,000 more than three times the amount in bonds. How much is each investment?

Solve Direct Translation Applications

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

Exercise 39

Pam is 3 years older than her sister, Jan. The sum of their ages is 99. Find their ages.

Answer

Pam is 51 and Jan is 48.

Exercise 40

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

Solve Geometry Applications

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

Exercise 41

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

Answer

The measures are 119 degrees and 61 degrees.

Exercise 42

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

Exercise 43

Becca is hanging a 28 foot floral garland on the two sides and top of a pergola to prepare for a wedding. The height is four feet less than the width. Find the height and width of the pergola.

Answer

The pergola is 8 feet high and 12 feet wide.

Exercise 44

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.

Solve Uniform Motion Applications

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

Exercise 45

Sheila and Lenore were driving to their grandmother’s house. Lenore left one hour after Sheila. Sheila drove at a rate of 45 mph, and Lenore drove at a rate of 60 mph. How long will it take for Lenore to catch up to Sheila?

Answer

It will take Lenore 3 hours.

Exercise 46

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 (34 hour) later, driving her car at a rate of 25 miles per hour. How long will it take Cheryl to catch up to Bob?

Exercise 47

Marcus can drive his boat 36 miles down the river in three hours but takes four hours to return upstream. Find the rate of the boat in still water and the rate of the current.

Answer

The rate of the boat is 10.5 mph. The rate of the current is 1.5 mph.

Exercise 48

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.

Solve Mixture Applications with Systems of Equations

Solve Mixture Applications

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

Exercise 49

Lynn paid a total of $2,780 for 261 tickets to the theater. Student tickets cost $10 and adult tickets cost $15. How many student tickets and how many adult tickets did Lynn buy?

Answer

Lynn bought 227 student tickets and 34 adult tickets.

Exercise 50

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?

Exercise 51

Yumi wants to make 12 cups of party mix using candies and nuts. Her budget requires the party mix to cost her $1.29 per cup. The candies are $2.49 per cup and the nuts are $0.69 per cup. How many cups of candies and how many cups of nuts should she use?

Answer

Yumi should use 4 cups of candies and 8 cups of nuts.

Exercise 52

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?

Solve Interest Applications

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

Exercise 53

Jack has $12,000 to invest and wants to earn 7.5% interest per year. He will put some of the money into a savings account that earns 4% per year and the rest into CD account that earns 9% per year. How much money should he put into each account?

Answer

Jack should put $3600 into savings and $8400 into the CD.

Exercise 54

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 $1585. What is the amount of each loan?

Graphing Systems of Linear Inequalities

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.

Exercise 55

{4x+y>63xy12

  1. (2,−1)
  2. (3,−2)
Answer
  1. yes
  2. no
Exercise 56

{y>13x+2x14y10

  1. (6,5)
  2. (15,8)

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

Exercise 57

{y<3x+1yx2

Answer

This figure shows a graph on an x y-coordinate plane of y is less than 3x + 1 and y is greater than or equal to -x - 2. The area to the right of each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.

Exercise 58

{xy>1y<13x2

Exercise 59

{2x3y<63x+4y12

Answer

This figure shows a graph on an x y-coordinate plane of 2x – 3y is less than 6 and 3x + 4y is greater than or equal to 12. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.

Exercise 60

{y34x+1x5

Exercise 61

{x+3y<5y13x+6

Answer

No solution

This figure shows a graph on an x y-coordinate plane of x + 3y is less than 5 and y is greater than or equal to -(1/3)x + 6. The area to the above or below each line is shaded different colors. There is no overlapping shaded area. One line is dotted.

Exercise 62

{y2x56x+3y>4

Solve Applications of Systems of Inequalities

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

Exercise 63

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.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Should she display 26 bracelets and 14 necklaces?
  4. Should she display 39 bracelets and 1 necklace?
Answer
  1. {b+n4012b+18n500
This figure shows a graph on an x y-coordinate plane of b + n is less than or equal to 40 and 12b + 18n is greater than or equal to 500. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.

3. yes
4. no

Exercise 64

Annie has a budget of $600 to purchase paperback books and hardcover books for her classroom. She wants the number of hardcover to be at least 5 more than three times the number of paperback books. Paperback books cost $4 each and hardcover books cost $15 each.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Can she buy 8 paperback books and 40 hardcover books?
  4. Can she buy 10 paperback books and 37 hardcover books?

Practice Test

Exercise 1

{x4y=82x+5y=10

  1. (0,2)
  2. (4,3)
Answer
  1. yes
  2. no

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

Exercise 2

{xy=5x+2y=4

Exercise 3

{xy>2y3x+1

Answer

This figure shows a graph on an x y-coordinate plane x – y is greater than -2 and y is less than or equal to 3x + 1. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.

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

Exercise 4

{3x2y=3y=2x1

Exercise 5

{x+y=3xy=11

Answer

(4,−7)

Exercise 6

{4x3y=75x2y=0

Exercise 7

{y=45x+18x+10y=10

Answer

infinitely many solutions

Exercise 8

{2x+3y=124x+6y=16

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

Exercise 9

The sum of two numbers is −24. One number is 104 less than the other. Find the numbers.

Answer

The numbers are 40 and 64

Exercise 10

Ramon wants to plant cucumbers and tomatoes in his garden. He has room for 16 plants, and he wants to plant three times as many cucumbers as tomatoes. How many cucumbers and how many tomatoes should he plant?

Exercise 11

Two angles are complementary. The measure of the larger angle is six more than twice the measure of the smaller angle. Find the measures of both angles.

Answer

The measures of the angles are 28 degrees and 62 degrees.

Exercise 12

On Monday, Lance ran for 30 minutes and swam for 20 minutes. His fitness app told him he had burned 610 calories. On Wednesday, the fitness app told him he burned 695 calories when he ran for 25 minutes and swam for 40 minutes. How many calories did he burn for one minute of running? How many calories did he burn for one minute of swimming?

Exercise 13

Kathy left home to walk to the mall, walking quickly at a rate of 4 miles per hour. Her sister Abby left home 15 minutes later and rode her bike to the mall at a rate of 10 miles per hour. How long will it take Abby to catch up to Kathy?

Answer

It will take Kathy 16 of an hour (or 10 minutes)

Exercise 14

It takes 512 hours for a jet to fly 2,475 miles with a headwind from San Jose, California to Lihue, Hawaii. The return flight from Lihue to San Jose with a tailwind, takes 5 hours. Find the speed of the jet in still air and the speed of the wind.

Exercise 15

Liz paid $160 for 28 tickets to take the Brownie troop to the science museum. Children’s tickets cost $5 and adult tickets cost $9. How many children’s tickets and how many adult tickets did Liz buy?

Answer

Liz bought 23 children’s tickets and 5 adult tickets.

Exercise 16

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?

Exercise 1

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.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Can she buy 20 candy bars and 70 lollipops?
  4. Can she buy 15 candy bars and 65 lollipops?
Answer
  1. {C+0.5L50L3C
This figure shows a graph on an x y-coordinate plane of C + 0.5L is less than or equal to 50 and L is greater than or equal to 3C. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.

3. No
4. Yes


This page titled Chapter 5 Review Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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