2. 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.

3. 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

4. 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

5. 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.

6. SKILL DEVELOPMENT / GUIDED PRACTICE1.
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?

7. 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

8. 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?

9. 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.

10. SKILL DEVELOPMENT / GUIDED PRACTICE1.
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?

11. SKILL DEVELOPMENT / GUIDED PRACTICE1.
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

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

13. 1
INDEPENDENT PRACTICE