1

Joe decides to invest in an arcade. Joe doesn't have a lot of capital, so he

makes a deal with the landlord to pay $400 for the first month and then to

increase the rent $10 per month for the first year. The first month

Joe earns $600, but when the kids figure out how to beat the game fewer

and fewer kids play it and each month Joe's revenue decreases by $30.

When will the money Joe pays in rent eq the money he earns from

kids playing the g

ual

ame?

2

Joe decides to invest in an arcade. Joe doesn't have a lot of capital, so he

makes a deal with the landlord to pay $400 for the first month and then to

increase the rent $10 per month for the first year. The first month

Joe earns $600, but when the kids figure out how to beat the game fewer

and fewer kids play it and each month Joe's revenue decreases by $30.

When will the money Joe pays in rent eq the money he earns from

kids playing the g

ual

ame?

Let’s organize our information!

4

MonthRent Revenue

0 $400.00 $600.00

1 $410.00 $570.00

2 $420.00 $540.00

Joe decides to invest in an arcade. Joe doesn't have a lot of capital, so he

makes a deal with the landlord to pay $400 for the first month and then to

increase the rent $10 per month for the first year. The first month

Joe earns $600, but when the kids figure out how to beat the game fewer

and fewer kids play it and each month Joe's revenue decreases by $30.

When will the money Joe pays in rent eq the money he earns from

kids playing the g

ual

ame?

5

Month Rent Revenue

0 $400.00 $600.00

1 $410.00 $570.00

2 $420.00 $540.00

3 $430.00 $510.00

4 $440.00 $480.00

5 $450.00 $450.00

6 $460.00 $420.00

7 $470.00 $390.00

8 $480.00 $360.00

9 $490.00 $330.00

10 $500.00 $300.00

11 $510.00 $270.00

12 $520.00 $240.00

Let’s look at the data in a graph form!

7

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

8

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

These are called a linear functions. Why?

Revenue

Rent

What do you call the place where two roads cross?

intersection

9

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

Revenue

Rent

Where does the revenue function begin?

Where does the rent function begin?

10

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

Revenue

Rent

By looking at the graph, how can you tell which linear function is steeper?

By looking at the table, how can you tell which linear function is steeper?

11

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

Revenue

Rent

By looking at the graph, how can you tell which linear function is decreasing?

By looking at the table, how can you tell which linear function is decreasing?

12

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

Revenue

Rent

By looking at the graph, how can you tell which linear function is increasing?

By looking at the table, how can you tell which linear function is increasing?

13

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

Revenue

Rent

How much does the rent function change each month?

How does the graph show this change?

How does the table show this change?

14

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenu

02 4 6 8 10 12

(5, 450)

Revenue

Rent

How much does the revenue function change each month?

How does the graph show this change?

How does the table show this change?

Let’s look at the informationin a symbolic/algebraic form!

16

Month Rent Revenue

0 $400.00 $600.00

1 $410.00 $570.00

2 $420.00 $540.00

3 $430.00 $510.00

4 $440.00 $480.00

5 $450.00 $450.00

6 $460.00 $420.00

7 $470.00 $390.00

8 $480.00 $360.00

9 $490.00 $330.00

10 $500.00 $300.00

11 $510.00 $270.00

12 $520.00 $240.00

400400 10

400 10 10 400 10 10 10

600600 30

600 30 30

600 30 30 30

17

Month Rent Revenue

0 $400.00 $600.00

1 $410.00 $570.00

2 $420.00 $540.00

3 $430.00 $510.00

4 $440.00 $480.00

5 $450.00 $450.00

6 $460.00 $420.00

7 $470.00 $390.00

8 $480.00 $360.00

9 $490.00 $330.00

10 $500.00 $300.00

11 $510.00 $270.00

12 $520.00 $240.00

400 0 10

400 1 10

400 2 10

400 3 10

600 0 30 600 1 30

600 2 30

600 3 30

400 4 10

400 5 10

400 6 10

400 7 10

400 8 10 400 9 10

400 10 10

400 11 10

400 12 10

600 4 30 600 5 30

600 6 30

600 7 30

600 8 30

600 9 30

600 10 30 600 11 30

600 12 30

Rent

WHAT’S THE PATTERN?

Rent 400 10t

Rent 400 10t

400 0 10

400 1 10

400 2 10

400 3 10

400 4 10

400 5 10

400 6 10

400 7 10

400 8 10 400 9 10

400 10 10

400 11 10

400 12 10

19

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

Rent

Rent 400 10t starting point

20

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

Rent Revenue

02 4 6 8 10 12

(5, 450)

Rent

Rent 400 10t

rate of change

Revenue

WHAT’S THE PATTERN?

Revenue 600 30t

Revenue 600 30t

600 0 30 600 1 30

600 2 30

600 3 30

600 4 30 600 5 30

600 6 30

600 7 30

600 8 30

600 9 30

600 10 30 600 11 30

600 12 30

22

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

Revenue

Revenue 600 30t starting point

rate of change

Let’s get abstract!

24

Rent 400 10t

y b mx

y mx b RentRevenue

y-axis

months

x-axis

Rent 400 10t

y mx b Revenue 600 30t

beginning amount

y when 0x "y intercept"

Rent 400 10t

y mx b Revenue 600 30t

rate of changechange in y over the change in x

"slope"

A B

y 5 x y 2 8x

C D

y 6x y 4x

E F

1y

3x

y 4 3x

30

Month Rent Revenue

0 $400 $600

1 $410 $570

2 $420 $540

3 $430 $510

4 $440 $480

5 $450 $450

6 $460 $420

7 $470 $390

8 $480 $360

9 $490 $330

10 $500 $300

11 $510 $270

12 $520 $240

$400

$450

$500

$550

$600

Months

RentRevenue

02 4 6 8 10 12

(5, 450)

Revenue

Rent

intersection

HOW CAN WE FIND THE INTERSECTION?Rent 400 10t Revenue 600 30t

Rent 400 10t Revenue 600 30t

400 10 600 30t t

10 30 600 400t t

40 200t 40 200

40 40t

5t

Rent 400 10t Revenue 600 30t

5t

Rent 400 0 51 Rent 400 50

Rent 450

Revenue 600 0 53

Revenue 600 150

Revenue 450

Let’s solve algebraic systemsof linear equations!

2 1

2 12

x y

x y

12 2x y

2 1y 12 2y

24 4 1y y

24 5 1y 5 1 24y 5 25y

5y

12 2x y 12 2x 5

2x

2, 5

2 1

2 12

x y

x y

1 2y x

2 12x 1 2x

2 4 12x x

2 5 12x 5 12 2x 5 10x

2x

1 2y x 1 2y 2

5y

2, 5

2 1

2 12

x y

x y

2, 5

2 1x y

2 1 2 5

2 12x y 2 12 2 5

2 1

2 12

x y

x y

Let’s graph this baby!

1 2y x 2 12y x

16

2y x

2 1

2 12

x y

x y

1 2y x 1

62

y x

2, 5

4 2 6

3 7 1

x y

x y

2 6 4

3 2

y x

y x

3 7 1x 3 2x

3 21 14 1x x

21 11 1x 11 1 21x 11 22x

2x

3 2y y 3 2y 2

3 4y

2, 11y

Let’s graph this baby!

7 1 3y x 1 3

7 7y x

4 2 6

3 7 1

x y

x y

3 2y x

5

4 10

x y

x y

5x y

4 10y 5y

5 4 10y y

3 5 10y 3 10 5y

3 5y 5

3y

5x y 5x 5 3

5 15

3 3x

10 5,

3 3

10

3x

5

4 10

x y

x y

Add the columns

0 3 5x y 3 5y

5

3y

5x y 5x 5 3

55

3x

15 5

3 3x

10

3x 10 5

,3 3

v

3 4

6 2 7

x y

x y

4 3y x

6 2 7x 4 3x6 8 6 7x x

8 7

WHAT!!??

3 4

6 2 7

x y

x y

3 4y x

y mx b

2 6 7y x 73

2y x

3 4

6 2 7

x y

x y

4 3y x 7

32

y x

3 2 10

6 4 20

x y

x y

6 4 20x y

6 4 20

6 4 20

x y

x y

Add the columns

0 0 0x y 0 0

True that but what does it mean?

3 2 10

6 4 20

x y

x y

They are the SAME Line!

2 10 3y x 4 20 6y x

35

2y x

65

4y x

35

2y x

Infinite Solutions

13

2

x y

x y

13y x

2x 13 x13 2x x

2 13 2x 2 2 13x 2 15x

15

2x

13y x 13y 15

226 15

2 2y

15 11,

2 2

Solve it by solving for y in the first equation first.

11

2y

13

2

x y

x y

2x y

13y 2 y

2 2 13y 2 13 2y

2 11y 11

2y

2x y 2x 11

2

4 11

2 2x

15 11,

2 2

Solve it by solving for x in the second equation first.

15

2x

13

2

x y

x y

2 0 15x y

2 15x 15

2x

13x y

13y 15

2

1513

2y

15 11,

2 2

Is there another way to solve this?

26 15 11

2 2 2y