Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 4 Systems of Linear Equations and Inequalities

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 Systems of Equations and Inequalities 4.1Solving Systems of Linear Equations in Two Variables 4.2Solving Systems of Linear Equations in Three Variables 4.3Solving Applications Using Systems of Equations CHAPTER 4

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 Solving Applications Using Systems of Equations 1.Solve application problems that translate to a system of two linear equations. 4.3

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Solving Problems Using Systems of Linear Equations To solve problems using a system of linear equations, 1. Select a variable to represent each unknown. 2. Write a system of equations. 3. Solve the system.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Example Mia sells concessions at a movie theatre. In one hour, she sells 78 popcorns. The popcorn sizes are small, which sell for $4 each, and large, which sell for $6 each. If her sales totaled $420, then how many of each size popcorn did she sell? Understand The unknowns are the number of each size popcorn. One relationship involves the number of popcorns (78 total), and the other relationship involves the total sales in dollars ($420).

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 continued Plan and Execute Let x represent the number of small popcorns. Let y represent the number of large popcorns. Relationship 1: The total number sold is 78. Translation: x + y = 78 Relationship 2: The total sales revenue is $420. Translation: 4x + 6y = 420 CategoryPriceNumberRevenue Small$4x4x4x Large$6y6y6y

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 continued Our system: Use elimination; choose x. Multiply by  4.  Substitute to find x.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 continued Answer Mia sold 54 large popcorns and 24 small popcorns. Check Verify both given relationships.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Example Anita and Ernesto are traveling north in separate cars on the same highway. Anita is traveling at 55 miles per hour and Ernesto is traveling at 70 miles per hour. Anita passes Exit 54 at 1:30 p.m. Ernesto passes the same exit at 1:45 p.m. At what time will Ernesto catch up with Anita? Understand To determine what time Ernesto will catch up with Anita, we need to calculate the amount of time it will take him to catch up to her. We can then add the amount to 1:45.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 continued Plan and Execute Let x = Anita’s travel time after passing the exit. Let y = Ernesto’s travel time after passing the exit. Relationship 1: Ernesto passes the exit 15 minutes after Anita; Anita will have traveled 15 minutes longer. Translation: x = y + ¼ (1/4 of an hour) Relationship 2: When Ernesto catches up, they will have traveled the same distance. Translation: 55x = 70y CategoryRateTimeDistance Anita55x55x Ernesto70y70y

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 continued Our system: Use substitution: Substitute to find x.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 continued Answer Ernesto will catch up to Anita in a little over 1 hour (1.17, which is 1 hour 10 minutes). The time will be 1:45 p.m. + 1 hr 10 minutes = 2:55 p.m. Check Verify both given relationships.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Example How many milliliters of a 20% HCl solution and 50% HCl solution must be mixed together to make 500 milliliters of 35% HCl solution? Understand The unknowns are the volumes of 20% and 50% solution that are mixed. One relationship involves the concentrations of each solution in the mixture and the other relationship involves the total volume of the final mixture (500 ml).

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 continued Plan and Execute Let x and y represent the two amounts to be mixed. Relationship 1: The total volume is 500 ml. Translation: x + y = 500 Relationship 2: The combined volumes of HCl in the two mixtures is to be 35% of the total mixture. Translation: 0.20x y = 0.35(500) SolutionConcentrationVolumeAmount of HCl 20%0.20x0.20x 50%0.50y0.50y 35%0.35x + y0.35(500)

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 continued Our system: Use elimination; choose x. Multiply by   Substitute to find x.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16 continued Answer Mixing 250 ml of 20% solution with 250 ml of 50% solution gives 500 ml of 35% solution. Check Verify both given relationships.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17 Example At a movie theatre, Kara buys one popcorn, two drinks and 2 candy bars, all for $12. Rebecca buys two popcorns, three drinks, and one candy bar for $17. Leah buys one popcorn, one drink and three candy bars for $11. Find the individual cost of one popcorn, one drink and one candy bar. Understand We have three unknowns and three relationships, and we are to find the cost of each. Plan Select a variable for each unknown, translate the relationship to a system of three equations, and then solve the system.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 18 continued Execute: p = popcorn, d = drink, and c = candy Relationship 1: one popcorn, two drinks and two candy bars, cost $12 Translation: p + 2d + 2c = 12 Relationship 2: two popcorns, three drinks, and one candy bar cost $17 Translation: 2p + 3d + c = 17 Relationship 3: one popcorn, one drink and three candy bars cost $11 Translation: p + d + 3c = 11

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 19 continued Our system: Choose to eliminate p: Start with equations 1 and 3. Choose to eliminate p: Start with equations 1 and 2.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 20 continued Use equations 4 and 5 to eliminate d. Substitute for c in d – c = 1 Substitute for c and d in p + d + 3c = 11 p (1.5) = 11 p + 7 = 11 p = 4 The cost of one candy bar is $1.50. The cost of one drink is $2.50. The cost of one popcorn is $4.00.