Download - AA Notes 5-2
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Section 5-2
Solving Systems Using Tables or Graphs
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Warm-up
Graph the following equations on the same graph.
a. y = 2/3x - 5
b. y = -x + 3
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Warm-up
Graph the following equations on the same graph.
a. y = 2/3x - 5
b. y = -x + 3x
y
y = 2/3x - 5
y = -x + 3
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How can we solve a system?
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System:
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System: Two sentences connected by “and”
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System: Two sentences connected by “and”
y = 4x + 8 and y = 3x + 5
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System: Two sentences connected by “and”
y = 4x + 8 and y = 3x + 5
Notation:
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System: Two sentences connected by “and”
y = 4x + 8 and y = 3x + 5
Notation:
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
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System: Two sentences connected by “and”
y = 4x + 8 and y = 3x + 5
Notation:
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
Solution Set of a System:
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System: Two sentences connected by “and”
y = 4x + 8 and y = 3x + 5
Notation:
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
Solution Set of a System: Since we’re using “and,” we are looking for what is in both sets/equations
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System: Two sentences connected by “and”
y = 4x + 8 and y = 3x + 5
Notation:
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
Solution Set of a System: Since we’re using “and,” we are looking for what is in both sets/equations
**Often, it is just a point**
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
0
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
0 8 5
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
0 8 5 3
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01
8 5 3
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01
8 512 8
3
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01
8 512 8
34
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01-2
8 512 8
34
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01-2
8 512 80 -1
34
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01-2
8 512 80 -1
341
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01-2-3
8 512 80 -1
341
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01-2-3
8 512 80 -1-4 -4
341
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01-2-3
8 512 80 -1-4 -4
3410
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01-2-3
8 512 80 -1-4 -4
3410
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
a. Tables x y = 4x + 8 y = 3x + 5
01-2-3
8 512 80 -1-4 -4
3410
(x, y) = (-3, 4)
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
c. Graphing Calculator
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
c. Graphing Calculator
1. Graph both equations
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
c. Graphing Calculator
1. Graph both equations
x
y
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
c. Graphing Calculator
1. Graph both equations
x
y
2. Press 2nd --> Calc
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
c. Graphing Calculator
1. Graph both equations
x
y
2. Press 2nd --> Calc3. Choose 5: intersect
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
c. Graphing Calculator
1. Graph both equations
x
y
2. Press 2nd --> Calc3. Choose 5: intersect
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
c. Graphing Calculator
1. Graph both equations
x
y
2. Press 2nd --> Calc3. Choose 5: intersect
4. Choose the two equations
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
c. Graphing Calculator
1. Graph both equations
x
y
2. Press 2nd --> Calc3. Choose 5: intersect
4. Choose the two equationsx
y
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Example 1
Solve the system
y = 4x + 8
y = 3x + 5
⎧⎨⎩⎪
b. Graphing (by hand)
c. Graphing Calculator
1. Graph both equations
x
y
2. Press 2nd --> Calc3. Choose 5: intersect
4. Choose the two equationsx
y
(x, y) = (-3, 4)
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Example 2
Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do
this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?
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Example 2
Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do
this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?
Barn l
w
w
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Example 2
Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do
this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?
Barn l
w
ww = width, l = length
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Example 2
Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do
this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?
Barn l
w
ww = width, l = length
l + 2w = 70
lw = 400
⎧⎨⎩⎪
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Example 2
Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do
this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?
Barn l
w
ww = width, l = length
l + 2w = 70
lw = 400
⎧⎨⎩⎪
Can we accurately solve this by graphing?
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Example 2
Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do
this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?
Barn l
w
ww = width, l = length
l + 2w = 70
lw = 400
⎧⎨⎩⎪
Can we accurately solve this by graphing?
(l, w) ≈ (14.4, 27.8)
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Example 2
Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do
this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?
Barn l
w
ww = width, l = length
l + 2w = 70
lw = 400
⎧⎨⎩⎪
Can we accurately solve this by graphing?
(l, w) ≈ (14.4, 27.8)(l, w) ≈ (55.6, 7.2)
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Example 2
Matt Mitarnowski wants to enclose a 400-square meter rectangular garden with 70 meters of fencing. To do
this, Matt must use one side of his barn as a side of the garden. What can the dimensions of the garden be?
Barn l
w
ww = width, l = length
l + 2w = 70
lw = 400
⎧⎨⎩⎪
Can we accurately solve this by graphing?
(l, w) ≈ (14.4, 27.8)(l, w) ≈ (55.6, 7.2)
14.4m x 27.8m or55.6m x 7.2 m
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
a. Table
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
0
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
0 0 1
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
0 0 1 1
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01
0 1 1
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01
0 15 -2
1
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01
0 15 -2
17
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1
0 15 -2
17
![Page 57: AA Notes 5-2](https://reader035.vdocuments.us/reader035/viewer/2022062419/558d43dcd8b42a4d318b45e9/html5/thumbnails/57.jpg)
Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1
0 15 -2-5 4
17
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1
0 15 -2-5 4
17-9
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1.1
0 15 -2-5 4
17-9
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1.1
0 15 -2-5 4.5 .7
17-9
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1.1
0 15 -2-5 4.5 .7
17-9.2
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1.1.2
0 15 -2-5 4.5 .7
17-9.2
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1.1.2
0 15 -2-5 4.5 .71 .4
17-9.2
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Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1.1.2
0 15 -2-5 4.5 .71 .4
17-9.2.6
![Page 65: AA Notes 5-2](https://reader035.vdocuments.us/reader035/viewer/2022062419/558d43dcd8b42a4d318b45e9/html5/thumbnails/65.jpg)
Example 3
Solve
y = 5x
y = −3x +1
⎧⎨⎩⎪
x y = 5x y = -3x + 1a. Table
01-1.1.2
0 15 -2-5 4.5 .71 .4
17-9.2.6
How long do we try this?
![Page 66: AA Notes 5-2](https://reader035.vdocuments.us/reader035/viewer/2022062419/558d43dcd8b42a4d318b45e9/html5/thumbnails/66.jpg)
Example 3
Solve
b. Graphing (by hand)
y = 5x
y = −3x +1
⎧⎨⎩⎪
![Page 67: AA Notes 5-2](https://reader035.vdocuments.us/reader035/viewer/2022062419/558d43dcd8b42a4d318b45e9/html5/thumbnails/67.jpg)
Example 3
Solve
b. Graphing (by hand)
y = 5x
y = −3x +1
⎧⎨⎩⎪
c. Graphing Calculator
![Page 68: AA Notes 5-2](https://reader035.vdocuments.us/reader035/viewer/2022062419/558d43dcd8b42a4d318b45e9/html5/thumbnails/68.jpg)
Example 3
Solve
b. Graphing (by hand)
y = 5x
y = −3x +1
⎧⎨⎩⎪
c. Graphing Calculatorx
y
![Page 69: AA Notes 5-2](https://reader035.vdocuments.us/reader035/viewer/2022062419/558d43dcd8b42a4d318b45e9/html5/thumbnails/69.jpg)
Example 3
Solve
b. Graphing (by hand)
y = 5x
y = −3x +1
⎧⎨⎩⎪
c. Graphing Calculatorx
y
x
y
![Page 70: AA Notes 5-2](https://reader035.vdocuments.us/reader035/viewer/2022062419/558d43dcd8b42a4d318b45e9/html5/thumbnails/70.jpg)
Example 3
Solve
b. Graphing (by hand)
y = 5x
y = −3x +1
⎧⎨⎩⎪
c. Graphing Calculatorx
y
x
y(x, y) = (.125, .625)
![Page 71: AA Notes 5-2](https://reader035.vdocuments.us/reader035/viewer/2022062419/558d43dcd8b42a4d318b45e9/html5/thumbnails/71.jpg)
Homework
![Page 72: AA Notes 5-2](https://reader035.vdocuments.us/reader035/viewer/2022062419/558d43dcd8b42a4d318b45e9/html5/thumbnails/72.jpg)
Homework
p. 282 #1-24
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