1. 2 4 month rent revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00
TRANSCRIPT
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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?
![Page 2: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/2.jpg)
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?
![Page 3: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/3.jpg)
Let’s organize our information!
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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?
![Page 5: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/5.jpg)
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
![Page 6: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/6.jpg)
Let’s look at the data in a graph form!
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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)
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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
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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?
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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?
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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?
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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?
![Page 13: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/13.jpg)
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?
![Page 14: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/14.jpg)
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?
![Page 15: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/15.jpg)
Let’s look at the informationin a symbolic/algebraic form!
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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
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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
![Page 18: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/18.jpg)
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
![Page 19: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/19.jpg)
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
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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
![Page 21: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/21.jpg)
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
![Page 22: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/22.jpg)
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
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Let’s get abstract!
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24
Rent 400 10t
y b mx
y mx b RentRevenue
y-axis
months
x-axis
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Rent 400 10t
y mx b Revenue 600 30t
beginning amount
y when 0x "y intercept"
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Rent 400 10t
y mx b Revenue 600 30t
rate of changechange in y over the change in x
"slope"
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A B
y 5 x y 2 8x
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C D
y 6x y 4x
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E F
1y
3x
y 4 3x
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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
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Rent 400 10t Revenue 600 30t
400 10 600 30t t
10 30 600 400t t
40 200t 40 200
40 40t
5t
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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
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Let’s solve algebraic systemsof linear equations!
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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
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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
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2 1
2 12
x y
x y
2, 5
2 1x y
2 1 2 5
2 12x y 2 12 2 5
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2 1
2 12
x y
x y
Let’s graph this baby!
1 2y x 2 12y x
16
2y x
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2 1
2 12
x y
x y
1 2y x 1
62
y x
2, 5
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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
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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
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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
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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
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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
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3 4
6 2 7
x y
x y
4 3y x 7
32
y x
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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?
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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
![Page 47: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/47.jpg)
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
![Page 48: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/48.jpg)
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
![Page 49: 1. 2 4 Month Rent Revenue 0 $400.00$600.00 1 $410.00$570.00 2 $420.00$540.00](https://reader030.vdocuments.us/reader030/viewer/2022032607/56649ec95503460f94bd65e9/html5/thumbnails/49.jpg)
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