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Section 5.5 Solve Mixture Applications with Systems of Equations

Solve Mixture Applications

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

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?

Lynn bought 227 student tickets and 34 adult tickets.

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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?

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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?

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

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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?

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Solve Interest Applications

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

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?

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

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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?

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Section 5.6 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.

{ 4 x + y > 6 3 x y 12

( 2 , −1 ) ( 3 , −2 )

yes no

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{ y > 1 3 x + 2 x 1 4 y 10

( 6 , 5 ) ( 15 , 8 )

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Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

{ y < 3 x + 1 y x 2

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.

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{ x y > −1 y < 1 3 x 2

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{ 2 x 3 y < 6 3 x + 4 y 12

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.

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{ y 3 4 x + 1 x −5

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{ x + 3 y < 5 y 1 3 x + 6

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.

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{ y 2 x 5 −6 x + 3 y > −4

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Solve Applications of Systems of Inequalities

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

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

  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?


{ b + n 40 12 b + 18 n 500

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.
yes
no

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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?
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Practice test

{ x 4 y = −8 2 x + 5 y = 10

( 0 , 2 ) ( 4 , 3 )

yes no

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

{ x y = 5 x + 2 y = −4

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{ x y > −2 y 3 x + 1

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.

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

{ 3 x 2 y = 3 y = 2 x 1

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{ x + y = −3 x y = 11

( 4 , −7 )

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{ 4 x 3 y = 7 5 x 2 y = 0

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{ y = 4 5 x + 1 8 x + 10 y = 10

infinitely many solutions

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{ 2 x + 3 y = 12 −4 x + 6 y = −16

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

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

The numbers are 40 and 64

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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?

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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.

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

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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?

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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?

It will take Kathy 1 6 of an hour (or 10 minutes).

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It takes 5 1 2 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.

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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?

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

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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?

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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?


{ C + 0.5 L 50 L 3 C

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.
No
Yes

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Practice Key Terms 1

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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