Applications of Systems of Linear Equations 4.3 Applications of Systems of Linear Equations Solve geometry problems by using two variables. Solve money problems by using two variables. Solve mixture problems by using two variables. Solve distance–rate–time problems by using two variables. Solve problems with three variables by using a system of three equations.

Applications of Systems of Linear Equations PROBLEM-SOLVING HINT When solving an applied problem using two variables, it is a good idea to pick letters that correspond to the descriptions of the unknown quantities. For example above, we could choose c to represent the number of citrons and w to represent the number of wood apples. Slide 4.3- 2

Applications of Systems of Linear Equations Solving an Applied Problem by Writing a System of Equations Step 1 Read the problem, several times if necessary. What information is given? What is to be found? This is often stated in the last sentence. Step 2 Assign variables to represent the unknown values. Use a sketch, diagram, or table, as needed. Step 3 Write a system of equations using the variable expressions. Step 4 Solve the system of equations. Step 5 State the answer to the problem. Label it appropriately. Does it seem reasonable? Step 6 Check the answer in the words of the original problem. Slide 4.3- 3

Solve geometry problems by using two variables. Objective 1 Solve geometry problems by using two variables. Slide 4.3- 4

CLASSROOM EXAMPLE 1a Finding the Dimensions of a rectangle The length of a rectangle is 7 inches greater than its width. The perimeter of the rectangle is 50 inches. Find the length and width of the rectangle. Slide 4.3- 5

CLASSROOM EXAMPLE 1b Finding the Dimensions of a Soccer Field A rectangular soccer field has perimeter 360 yd. Its length is 20 yd more than its width. What are its dimensions? Slide 4.3- 6

x 3x + 10 y Finding the Dimensions of a the angles CLASSROOM EXAMPLE 1c Finding the Dimensions of a the angles x 3x + 10 y Slide 4.3- 7

Solve money problems by using two variables. Objective 2 Solve money problems by using two variables. Slide 4.3- 8

(Source: Team Marketing Report.) CLASSROOM EXAMPLE 2 Solving a Problem about Ticket Prices For the 2009 Major League Baseball and National Football League seasons, based on average ticket prices, three baseball tickets and two football tickets would have cost $229.90. Two baseball tickets and one football ticket would have cost $128.27. What were the average ticket prices for the tickets for the two sports? (Source: Team Marketing Report.) Slide 4.3- 9

CLASSROOM EXAMPLE 2 Solving a Problem about Investment Jack invested $40,000, part at 4% and the rest at 6%. The annual return is $1740. How much money is invested at each rate? Slide 4.3- 10

CLASSROOM EXAMPLE 2 Solving a Problem about Ticket Prices A total of $3000 is invested, part at 2% simple interest and part at 4%. If the total annual return from the two investments is $100, how much is invested in each rate? Slide 4.3- 11

Solve mixture problems by using two variables. Objective 3 Solve mixture problems by using two variables. Slide 4.3- 12

CLASSROOM EXAMPLE 3a Solving a Mixture Problem A grocer has some $4-per-lb coffee and some $8-per-lb coffee that she will mix to make 50 lb of $5.60-per-lb coffee. How many pounds of each should be used? Slide 4.3- 13

CLASSROOM EXAMPLE 3b Solving a Mixture Problem How many ounces of each of 5% hydrochloric acid and 20% hydrochloric acid must be combined to get 10 oz of solution that is 12.5% of hydrochloric acid? Slide 4.3- 14

CLASSROOM EXAMPLE 3c Solving a Mixture Problem How many gallons of each of 25% alcohol and 35% alcohol should be mixed to get 20 gallons of 32% alcohol? Slide 4.3- 15

Solve distance-rate-time problems by using two variables. Objective 4 Solve distance-rate-time problems by using two variables. Slide 4.3- 16

CLASSROOM EXAMPLE 4a Solving a Motion Problem A train travels 600 mi in the same time that a truck travels 520 mi. Find the speed of each vehicle if the train’s average speed is 8 mph faster than the truck’s. Slide 4.3- 17

CLASSROOM EXAMPLE 4b Solving a Motion Problem Jack can drive 225 miles in the same time that Matthew can drive 195 miles. If Jack’s speed is 10 mph faster than Matthew’s, find the speed of each of their vehicles. Slide 4.3- 18

CLASSROOM EXAMPLE 4c Solving a Motion Problem In his motorboat, Bill travels upstream at top speed to his favorite fishing spot, a distance of 36 mi, in 2 hr. Returning, he finds that the trip downstream, still at top speed, takes only 1.5 hr. Find the speed of Bill’s boat and the speed of the current. Slide 4.3- 19

CLASSROOM EXAMPLE 4d Solving a Motion Problem Traveling for 3 hr into a steady head wind, a plane flies 1650 mi. The pilot determines that flaying with the same wind for 2 hr, he could make a trip of 1300 mi. Find the speed of the plane and the speed of the wind. Slide 4.3- 20

Objective 5 Solve problems with three variables by using a system of three equations. Slide 4.3- 21

Solve problems with three variables by using a system of three equations. PROBLEM-SOLVING HINT If an application requires finding three unknown quantities, we can use a system of three equations to solve it. We extend the method used for two unknowns. Slide 4.3- 22

CLASSROOM EXAMPLE 5 Solving a Problem Involving Prices A department store display features three kinds of perfume: Felice, Vivid, and Joy. There are 10 more bottles of Felice than Vivid, and 3 fewer bottles of Joy than Vivid. Each bottle of Felice costs $8, Vivid costs $15, and Joy costs $32. The total value of the all the perfume is $589. How many bottles of each are there? Slide 4.3- 23

A paper mill makes newsprint, bond, and copy machine paper. CLASSROOM EXAMPLE 6 Solving a Business Production Problem A paper mill makes newsprint, bond, and copy machine paper. Each ton of newsprint requires 3 tons of recycled paper and 1 ton of wood pulp. Each ton on bond requires 2 tons of recycled paper, and 4 tons of wood pulp, and 3 tons of rags. Each ton of copy machine paper requires 2 tons of recycled paper, 3 tons of wood pulp, and 2 tons of rags. The mill has 4200 tons of recycled paper, 5800 tons of wood pulp, and 3900 tons of rags. How much of each kind of paper can be made from these supplies? Slide 4.3- 24