We will learn how to model1 with Linear Systems2. LEARNING OBJECTIVE CFU What are we going to learn today? Declare the Objective A: Read the Objective to B. B: Define Systems and Inequalities to A Definition Represents real-world situations and use them to solve problems. A set of two or more equations with same variables.

x + y = 19 x + y = 12 Solve the system. Lisa spends part of her year as a member of a gym. She then finds a better deal at another gym, so she cancels her membership with the first gym after x months and spends the rest of the year, y months, with the second gym. Write the equation: 1. A company has to buy computers, x, and printers, y. They buy total of 19 machine. Write the equation: 2 Make the Connection Students, you already know how to represent the equation using variables. Now, we are going to learn how to model with Linear Systems. x + y = 12 x + y = 19

Check for Understanding A PE teacher needs basketballs and volleyballs. She bought a total of 40 balls. Basketballs cost $7 each and volleyballs cost $5 each with a total spent of $230. Which linear system models the situation? How do you know? b + v = 40 7b + 5v = 230 B. b + v = 230 7b + 5v = 40 A linear system can be used to model a real world problem when two equations can be created. Angie’s Awesome cupcakes sells designer cupcakes. Jagpal bought some chocolate and some vanilla for a total of 20 cupcakes. Chocolate cupcakes cost $5 and vanilla cupcakes cost $3. It was $84 for the 2o cupcakes. The first equation is for the total number of machines. {TOTAL} The second equation is for the total cost of the machines. {COST} x + y = 20 5x + 3y = 20

Write a system of equations to represent the situation, then solve the system A company has to buy computers and printers. Each computer, x, costs $595 and each printer, y, costs $370. If the company spends $9,055 and buys a total of 19 machines, how many of each did it buy? A company has to buy computers and printers. Each computer, x, costs $525 and each printer, y, costs $335. If the company spends $7,125 and buys a total of 19 machines, how many of each did it buy? 595x + 370y = 9,055 x + y = 19 CONCEPT DEVELOPMENT 595x + 370y = 9,055 + −595x + −595y = −11,305 *Multiply by −525 −225y = −2,250 y = 10 x + y = 19 x + 10 = 19 x = 9 So, the company bought 9 computers and 10 printers. So, the company bought 4 computers and 15 printers.

SKILL DEVELOPMENT / GUIDED PRACTICE 1. Lisa spends part of her year as a member of a gym. She then finds a better deal at another gym, so she cancels her membership with the first gym after x months and spends the rest of the year, y months, with the second gym. The membership to the first gym cost $65, while the membership for the second gym cost $50. She ended up spending a total of $720 over the course of the year. Write the system only: Write the system of equations only Jack is selling tickets to an event. Attendees can either buy a general admission ticket, x, or a VIP ticket, y. The general admission tickets are $60 and the VIP tickets are $75. He does not know how many of each type he has sold, but he knows he sold a total of 29 tickets and made $1,965. Write the system only: Number of months x + y = 12 SKILL DEVELOPMENT / GUIDED PRACTICE Cost 65x + 50y = $720 Check for Understanding B Explain to A: How do you write the system of equations?

SKILL DEVELOPMENT / GUIDED PRACTICE A student has to buy graph paper and printer paper. The graph paper, x, costs $4, while the printer paper, y, costs $3. She wants to buy at least 5 packs of paper but wants to spend at most $17. Write the system: 1. Anthony is buying towels for his apartment. He finds some green towels, x, that cost $5 each and blue towels, y, that cost $8 each. He wants to buy at least 5 towels, but does not want to spend more than $41. How many of each towel can he purchase? Write the system: at least ≥ at most ≤ SKILL DEVELOPMENT / GUIDED PRACTICE The student may purchase 5 or more packs of paper. x + y ≥ 5 $17 is the spending limit, so the cost must be less than or equal to it. Angelique may purchase 5 or more towels. x + y ≥ 5 $4x + $3y ≤ 17 $41 is the spending limit, so the cost must be less than or equal to it. Check for Understanding B Explain to A: How do you write the systems of inequalities? 5x + 8y ≤ 41

SKILL DEVELOPMENT / GUIDED PRACTICE Check for Understanding A: How do you determine what side to shade? B: How do you determine the final solution area? 1 Solve the inequality for y (y = mx + b). 2 Graph the boundary line for the inequality. (< > , £, ³ ). 3 Shade the region (>, ³: Above, <, £: Below). 4 Repeat steps 1-3 for the second inequality. 5 Shade the region that overlap. 1. Graph of the system of inequalities. Remember the Concept y-Intercept (0,b) = . . Slope ( 𝑹𝒊𝒔𝒆 𝑹𝒖𝒏 ) = SKILL DEVELOPMENT / GUIDED PRACTICE overlap. + . . - Check for Understanding B Explain to A: How do you graph the systems of inequalities?

SKILL DEVELOPMENT / GUIDED PRACTICE Check for Understanding A: How do you determine what side to shade? B: How do you decide where to do the final shade? 1 Solve the inequality for y (y = mx + b). 2 Graph the boundary line for the inequality. (< > , £, ³ ). 3 Shade the region (>, ³: Above, <, £: Below). 4 Repeat steps 1-3 for the second inequality. 5 Shade the region that overlap. 1. Graph of the system of inequalities. 2 Graph of the system of inequalities. Remember the Concept SKILL DEVELOPMENT / GUIDED PRACTICE overlap. overlap.

SKILL DEVELOPMENT / GUIDED PRACTICE 1. Lisa spends part of her year as a member of a gym. She then finds a better deal at another gym, so she cancels her membership with the first gym after x months and spends the rest of the year, y months, with the second gym. The membership to the first gym cost $65, while the membership for the second gym cost $50. She ended up spending a total of $720 over the course of the year. Write and Solve system: Jack is selling tickets to an event. Attendees can either buy a general admission ticket, x, or a VIP ticket, y. The general admission tickets are $60 and the VIP tickets are $75. He does not know how many of each type he has sold, but he knows he sold a total of 29 tickets and made $1,965. Write and Solve the system: Number of months x + y = 12 SKILL DEVELOPMENT / GUIDED PRACTICE Cost 65x + 50y = $720 Check for Understanding B Explain to A: How do you write the system of equations?

SKILL DEVELOPMENT / GUIDED PRACTICE 1. A student has to buy graph paper and printer paper. The graph paper, x, costs $4, while the printer paper, y, costs $3. She wants to buy at least 5 packs of paper but wants to spend at most $17. Write & Solve the system: Anthony is buying towels for his apartment. He finds some green towels, x, that cost $5 each and blue towels, y, that cost $8 each. He wants to buy at least 5 towels, but does not want to spend more than $41. How many of each towel can he purchase? Write & Solve the system: at least ≥ at most ≤ SKILL DEVELOPMENT / GUIDED PRACTICE The student may purchase 5 or more packs of paper. x + y ≥ 5 $17 is the spending limit, so the cost must be less than or equal to it. Angelique may purchase 5 or more towels. x + y ≥ 5 $4x + $3y ≤ 17 $41 is the spending limit, so the cost must be less than or equal to it. Check for Understanding B Explain to A: How do you write the systems of inequalities? 5x + 8y ≤ 41

What did you learn today about how to model with Linear Systems Word Bank SUMMARY CLOSURE

1 INDEPENDENT PRACTICE