Int Math 2 Section 8-3 1011
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Transcript of Int Math 2 Section 8-3 1011
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Section 8-3Solve Systems by Substitution
Thursday, April 7, 2011
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Essential Question
How do you solve systems of equations using substitution?
Where you’ll see this:
Transportation, construction, sports, recreation
Thursday, April 7, 2011
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Substitution
Thursday, April 7, 2011
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Substitution
Plugging in a number or expression for a variable
Thursday, April 7, 2011
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Example 1
Solve when x = 3.
y = 3x - 2
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Example 1
Solve when x = 3.
y = 3x - 2
y = 3(3) - 2
Thursday, April 7, 2011
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Example 1
Solve when x = 3.
y = 3x - 2
y = 3(3) - 2
y = 9 - 2
Thursday, April 7, 2011
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Example 1
Solve when x = 3.
y = 3x - 2
y = 3(3) - 2
y = 9 - 2
y = 7
Thursday, April 7, 2011
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Solve for y
When substituting, we might need to rewrite an equation. To do this, we need to decide which equations to rewrite.
Thursday, April 7, 2011
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Solve for y
When substituting, we might need to rewrite an equation. To do this, we need to decide which equations to rewrite.
Which equation requires the fewest steps to solve for y?
3x −7 y =21 8x − y =16
4x + y =3
23
x 2 +5 y =9x −3 y x + y
2=3
Thursday, April 7, 2011
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Solve a System of Equations by Substitution
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Solve a System of Equations by Substitution
1. Solve one equation for one variable (your choice)
Thursday, April 7, 2011
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Solve a System of Equations by Substitution
1. Solve one equation for one variable (your choice)
2. Substitute the expression from the equation into the other equation
Thursday, April 7, 2011
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Solve a System of Equations by Substitution
1. Solve one equation for one variable (your choice)
2. Substitute the expression from the equation into the other equation
3. Solve for the variable and substitute back into the original equation to find the other variable
Thursday, April 7, 2011
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Solve a System of Equations by Substitution
1. Solve one equation for one variable (your choice)
2. Substitute the expression from the equation into the other equation
3. Solve for the variable and substitute back into the original equation to find the other variable
4. Rewrite your answer as an ordered pair and check it!
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =30
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =3010 10
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =3010 10 x =3
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =3010 10 x =3
2(3)− y =1
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =3010 10 x =3
2(3)− y =1 6− y =1
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =3010 10 x =3
2(3)− y =1 6− y =1 y =5
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =3010 10 x =3
2(3)− y =1 6− y =1 y =5Check:
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =3010 10 x =3
2(3)− y =1 6− y =1 y =5Check:
4(3)+3(5)=27
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =3010 10 x =3
2(3)− y =1 6− y =1 y =5Check:
4(3)+3(5)=27 2(3)−5=1
Thursday, April 7, 2011
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Example 2Solve the system of equations and check.
4x +3 y =272x − y =1⎧⎨⎪
⎩⎪
2x − y =1 −2x −2x
− y = −2x +1 y =2x −1
4x +3(2x −1)=27 4x +6x −3=27
10x −3=27 +3 +3
10x =3010 10 x =3
2(3)− y =1 6− y =1 y =5Check:
4(3)+3(5)=27 2(3)−5=1
(3, 5)Thursday, April 7, 2011
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Example 3In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
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Example 3In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.
Thursday, April 7, 2011
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Example 3In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.The number will be 10x + y.
Thursday, April 7, 2011
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Example 3In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.The number will be 10x + y.
Set up the system:
Thursday, April 7, 2011
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Example 3In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.The number will be 10x + y.
Set up the system:
x + y =9
Thursday, April 7, 2011
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Example 3In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.The number will be 10x + y.
Set up the system:
x + y =9 10x + y =12x
Thursday, April 7, 2011
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Example 3In a two-digit number, the sum of the digits is 9. The number is 12 times
the tens digit. Find the number using a system of equations.
Let x = tens digit and y = ones digit.The number will be 10x + y.
Set up the system:
x + y =9 10x + y =12x{
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
3+ y =9
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
3+ y =9
y =6
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
3+ y =9
y =6
Check:
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
3+ y =9
y =6
Check:
3+6=9
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
3+ y =9
y =6
Check:
3+6=9
10(3)+6=12(3)
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
3+ y =9
y =6
Check:
3+6=9
10(3)+6=12(3) 30+6=36
Thursday, April 7, 2011
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Example 3
x + y =910x + y =12x⎧⎨⎪
⎩⎪
y = −x +9
10x + (−x +9)=12x 9x +9=12x
9=3x x =3
3+ y =9
y =6
Check:
3+6=9
10(3)+6=12(3) 30+6=36
(3, 6)
Thursday, April 7, 2011
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Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
Thursday, April 7, 2011
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Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
Thursday, April 7, 2011
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Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
Thursday, April 7, 2011
![Page 58: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/58.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
Thursday, April 7, 2011
![Page 59: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/59.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
−14= −14
Thursday, April 7, 2011
![Page 60: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/60.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
−14= −14What’s going on here?
Thursday, April 7, 2011
![Page 61: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/61.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
−14= −14What’s going on here?
−2 y = −x +7
Thursday, April 7, 2011
![Page 62: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/62.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
−14= −14What’s going on here?
−2 y = −x +7
y =
12
x − 72
Thursday, April 7, 2011
![Page 63: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/63.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
−14= −14What’s going on here?
−2 y = −x +7
y =
12
x − 72
4 y =2x −14
Thursday, April 7, 2011
![Page 64: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/64.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
−14= −14What’s going on here?
−2 y = −x +7
y =
12
x − 72
4 y =2x −14
y =
12
x − 72
Thursday, April 7, 2011
![Page 65: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/65.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
−14= −14What’s going on here?
−2 y = −x +7
y =
12
x − 72
4 y =2x −14
y =
12
x − 72
These are the same lines!
Thursday, April 7, 2011
![Page 66: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/66.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
−14= −14What’s going on here?
−2 y = −x +7
y =
12
x − 72
4 y =2x −14
y =
12
x − 72
These are the same lines!
Infinitely many solutions on the line.
Thursday, April 7, 2011
![Page 67: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/67.jpg)
Example 4Solve each system of equations. Check your solution.
a. x −2 y =7
−2x +4 y = −14⎧⎨⎪
⎩⎪
x =2 y +7
−2(2 y +7)+4 y = −14
−4 y −14+4 y = −14
−14= −14What’s going on here?
−2 y = −x +7
y =
12
x − 72
4 y =2x −14
y =
12
x − 72
These are the same lines!
Infinitely many solutions on the line.
Thursday, April 7, 2011
![Page 68: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/68.jpg)
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
Thursday, April 7, 2011
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Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
Thursday, April 7, 2011
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Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
Thursday, April 7, 2011
![Page 71: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/71.jpg)
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
−4x +14 2
7x − 2
7⎛⎝⎜
⎞⎠⎟=3
Thursday, April 7, 2011
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Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
−4x +14 2
7x − 2
7⎛⎝⎜
⎞⎠⎟=3
−4x +4x −4=3
Thursday, April 7, 2011
![Page 73: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/73.jpg)
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
−4x +14 2
7x − 2
7⎛⎝⎜
⎞⎠⎟=3
−4x +4x −4=3 −4=3
Thursday, April 7, 2011
![Page 74: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/74.jpg)
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
−4x +14 2
7x − 2
7⎛⎝⎜
⎞⎠⎟=3
−4x +4x −4=3 −4=3
14 y = 4x +3
Thursday, April 7, 2011
![Page 75: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/75.jpg)
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
−4x +14 2
7x − 2
7⎛⎝⎜
⎞⎠⎟=3
−4x +4x −4=3 −4=3
14 y = 4x +3
y =
27
x + 314
Thursday, April 7, 2011
![Page 76: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/76.jpg)
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
−4x +14 2
7x − 2
7⎛⎝⎜
⎞⎠⎟=3
−4x +4x −4=3 −4=3
14 y = 4x +3
y =
27
x + 314
These lines are parallel.
Thursday, April 7, 2011
![Page 77: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/77.jpg)
Example 4Solve each system of equations. Check your solution.
b. 2x −7 y = −2
−4x +14 y =3⎧⎨⎪
⎩⎪
−7 y = −2x −2
y =
27
x − 27
−4x +14 2
7x − 2
7⎛⎝⎜
⎞⎠⎟=3
−4x +4x −4=3 −4=3
14 y = 4x +3
y =
27
x + 314
These lines are parallel.There are no solutions.
Thursday, April 7, 2011
![Page 78: Int Math 2 Section 8-3 1011](https://reader035.fdocuments.in/reader035/viewer/2022062308/559aa16a1a28abf1088b4793/html5/thumbnails/78.jpg)
Problem Set
p. 346 #1-29 odd
“I have high expectations for you, so I will wait a little longer. I know you can get it if I give you a chance.” - Myra and David Sadker
Thursday, April 7, 2011