Lesson 4 7 Core Focus on Linear Equations

Lesson 4. 7 Core Focus on Linear Equations Applications of Systems of Equations

Warm-Up Solve the system of equations using any method. y = 3 x + 4 − 2 x + 5 y = 20 (0, 4)

Lesson 4. 7 Applications of Systems of Equations Set up and solve systems of equations from word problems.

Good to Know! Systems of linear equations are used to solve problems in all types of realworld situations. You will be given details about a problem that will provide you with enough information to write two linear equations. Solving Systems of Equations from Word Problems 1. Carefully read the word problem and identify the important information. 2. Determine what each variable represents. 3. Write two linear equations that represent the situation in the word problem. 4. Determine which method you want to use to solve the problem and solve the system. 5. Always remember to check your answer by referring back to the original problem to see if your solution is correct.

Explore! At The Movies The Rodriguez family and the Jacobson family go to the movies together. The Rodriguez family bought 3 adult tickets and 2 youth tickets for a total of $29. 00. The Jacobson family bought 2 adult tickets and 5 youth tickets for a total of $31. 25. Let x represent the cost of an adult ticket and y represent the cost of a youth ticket. Step 1 Write an equation to represent the Rodriguez family’s movie ticket purchase. Step 2 Write an equation to represent the Jacobson family’s movie ticket purchase. Step 3 Choose the best method for solving this system of equations. Why did you choose that method? Step 4 Solve your system of linear equations.

Explore! At The Movies The Rodriguez family and the Jacobson family go to the movies together. The Rodriguez family bought 3 adult tickets and 2 youth tickets for a total of $29. 00. The Jacobson family bought 2 adult tickets and 5 youth tickets for a total of $31. 25. Let x represent the cost of an adult ticket and y represent the cost of a youth ticket. Step 5 How much did a youth’s ticket cost at this movie theater? What was the cost of an adult ticket? Step 6 Check your answer by determining if your ticket prices give the same totals that were charged to the Jacobson and Rodriguez family. Step 7 The Chang family also went to see the same movie as the other two families. The Changs bought one adult ticket and 3 youth’s tickets. What was the total cost for the Chang family to go to the movies?

Example 1 Nai is trying to decide between two different cell phone plans. Plan A charges a flat fee of $22 per month plus $0. 10 per minute of phone usage. Plan B charges $0. 18 per minute with no flat fee. a. How many minutes would Nai have to use each month for the cell phone plans to cost the same amount? How much would it cost? Let x represent the minutes talked and y represent the total monthly cost. Write a system of two linear equations to model this situation. Plan A: y = 22 + 0. 10 x Start value of $22 plus $0. 10 per minute. Plan B: y = 0. 18 x $0. 18 per minute Choose a method and solve the system of equations. Graphing or substitution would work. SUBSTITUTION will work best because it will provide an accurate answer since it is not known if the solution will have integer values.

Example 1 Continued… Nai is trying to decide between two different cell phone plans. Plan A charges a flat fee of $22 per month plus $0. 10 per minute of phone usage. Plan B charges $0. 18 per minute with no flat fee. a. How many minutes would Nai have to use each month for the cell phone plans to cost the same amount? How much would they cost? Solve the system. Substitute 0. 18 x for the y-variable. Substitute the x-value into one of the original equations to determine the total cost when the plans cost the same. y 0. 18 x – 0. 10 x 0. 08 x = 22 + 0. 10 x – 0. 10 x = 22 0. 08 = 275 minutes y = 0. 18(275) = $49. 50 The plans would cost the same amount, $49. 50, after 275 minutes.

Example 1 Continued… Nai is trying to decide between two different cell phone plans. Plan A charges a flat fee of $22 per month plus $0. 10 per minute of phone usage. Plan B charges $0. 18 per minute with no flat fee. b. Nai plans to talk 400 minutes each month. To determine which plan is best, substitute 400 for x in each equation to see which plan will be less expensive. Plan A: y = 22 + 0. 10(400) = $62 Plan B: y = 0. 18(400) = $72 Nai should choose Plan A if he plans to use his phone 400 minutes each month.

Example 2 Omar has two possible sales job options. Job Option #1 has a monthly salary of $1, 200 plus 4% of his total sales. Job Option #2 has a monthly salary of $1, 500 plus 2% of his total sales. a. How much would Omar have to sell to earn the same amount in one month at each job? Let x represent the amount of Omar’s sales in one month and y represent the total monthly salary. Job Option #1: y = 1200 + 0. 04 x 4% = 0. 04 Job Option #2: y = 1500 + 0. 02 x 2% = 0. 02 Choose a method and solve the system of equations. Graphing or substitution would work. SUBSTITUTION will work best because the y-intercepts are quite large and would be difficult to graph accurately.

Example 2 Continued… Omar has two possible sales job options. Job Option #1 has a monthly salary of $1, 200 plus 4% of his total sales. Job Option #2 has a monthly salary of $1, 500 plus 2% of his total sales. a. How much would Omar have to sell to earn the same amount in one month at each job? Solve the system. Substitute 1200 + 0. 04 x for y. Substitute the x-value into one of the original equations to determine the total salary when the jobs would pay the same. y = 1200 + 0. 04(15000) = $1800 y 1200 + 0. 04 x – 0. 02 x 1200 + 0. 02 x – 1200 0. 02 x 0. 02 x = 1500 + 0. 02 x – 0. 02 x = 1500 a – 1200 a = 300 a 0. 02 a a = $15, 000 a Omar will earn the same monthly income ($1, 800) at Job Option #1 or #2 if he sells $15, 000 worth of merchandise.

Example 2 Continued… Omar has two possible sales job options. Job Option #1 has a monthly salary of $1, 200 plus 4% of his total sales. Job Option #2 has a monthly salary of $1, 500 plus 2% of his total sales. b. Omar thinks he can sell an average of $8, 000 worth of merchandise in one month. Which job should he take? Use the original equations to determine which job will have the highest pay if he sells $8, 000 worth of merchandise. Remember that his sales amount is substituted for x. Job Option #1: y = 1200 + 0. 04(8000) = $1, 520 Job Option #2: y = 1500 + 0. 02(8000) = $1, 660 If Omar sells $8000 worth of merchandise per month, he should take Job Option #2.

Example 3 Sunshine Flowery Company (SFC) ships boxes of tulip bulbs. The bulbs are always shipped in boxes that are the exact same size and with. The billing statements are mailed in a separate envelope. On Monday, SFC shipped 210 boxes of bulbs and 140 billing statements. The total shipping bill for the day was $702. 10. On Tuesday, SFC shipped 70 boxes of bulbs and 80 billing statements. The total shipping bill for Tuesday was $243. 70. Determine the individual cost for mailing a box of bulbs and the cost for mailing a billing statement. Let x represent the cost of shipping a box of bulbs and y represent the cost of mailing a billing statement. Write a system of two linear equations to model this situation. Monday: 210 x + 140 y = 702. 10 Tuesday: 70 x + 80 y = 243. 70

Example 3 Continued… Determine the individual cost for mailing a box of bulbs and the cost for mailing a billing statement. Choose a method and solve the system of equations. This system of equations is set up for using the ELIMINATION method because the x- and y-variables are lined up in columns on one side of the equals sign and the constants are on the other side. Use multiplication in order to get opposite amounts of one variable. 210 x + 140 y = 702. 10 a 70 x + 80 y = 243. 70 – 3(70 x + 80 y = 243. 70) – 210 x – 240 y = – 731. 10 Solve the system by first getting one variable with opposite coefficients. Add the columns together and solve for y. Each billing statement costs $0. 29 to mail. 210 x + 140 y – 210 x – 240 y – 100 y = 702. 10 = – 731. 10 = – 29 a – 100 a = $0. 29 a

Example 3 Continued… Determine the individual cost for mailing a box of bulbs and the cost for mailing a billing statement. Substitute the y-value into one of the original equations to solve for x. Each box of bulbs costs $3. 15 to ship. 70 x + 80(0. 29) 70 x + 23. 20 – 23. 20 70 x 70 x = 243. 70 – 23. 20 = 220. 50 70 a = $3. 15 a

Communication Prompt What is most difficult for you when solving systems of linear equations application situation?

Exit Problem Jill begins the summer with $500 in her savings account. Each week, she withdraws $20. Her brother, Sam, starts the summer with $150 in his account. He adds $30 to his account each week. At what week will both Jill and Sam have the same amount in their account? Week 7