Int Math 2 Section 8-3 1011

Section 8-3Solve Systems by Substitution

Thursday, April 7, 2011

Essential Question

How do you solve systems of equations using substitution?

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Substitution


Substitution

Plugging in a number or expression for a variable


Example 1

Solve when x = 3.

y = 3x - 2


Example 1

Solve when x = 3.

y = 3x - 2

y = 3(3) - 2


Example 1

Solve when x = 3.

y = 3x - 2

y = 3(3) - 2

y = 9 - 2


Example 1

Solve when x = 3.

y = 3x - 2

y = 3(3) - 2

y = 9 - 2

y = 7


Solve for y

When substituting, we might need to rewrite an equation. To do this, we need to decide which equations to rewrite.


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


http://www.polleverywhere.com/multiple_choice_polls/OTEzNTcxOTEx

http://www.polleverywhere.com/multiple_choice_polls/OTEzNTcxOTEx

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





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!


Example 2Solve the system of equations and check.

4x +3 y =272x − y =1⎧⎨⎪

⎩⎪



4x +3 y =272x − y =1⎧⎨⎪

⎩⎪

2x − y =1



4x +3 y =272x − y =1⎧⎨⎪

⎩⎪

2x − y =1 −2x −2x



4x +3 y =272x − y =1⎧⎨⎪

⎩⎪

2x − y =1 −2x −2x

− y = −2x +1



4x +3 y =272x − y =1⎧⎨⎪

⎩⎪

2x − y =1 −2x −2x

− y = −2x +1 y =2x −1



4x +3 y =272x − y =1⎧⎨⎪

⎩⎪

2x − y =1 −2x −2x

− y = −2x +1 y =2x −1

4x +3(2x −1)=27



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



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



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



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



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



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



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



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



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



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:



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



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



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

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.




Let x = tens digit and y = ones digit.The number will be 10x + y.





Set up the system:





Set up the system:

x + y =9





Set up the system:

x + y =9 10x + y =12x





Set up the system:

x + y =9 10x + y =12x{


Example 3

x + y =910x + y =12x⎧⎨⎪

⎩⎪


Example 3

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9


Example 3

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x


Example 3

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x


Example 3

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x


Example 3

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3


Example 3

x + y =910x + y =12x⎧⎨⎪

⎩⎪

y = −x +9

10x + (−x +9)=12x 9x +9=12x

9=3x x =3

3+ y =9


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


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:


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


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)


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


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)


Example 4Solve each system of equations. Check your solution.

a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7



a. x −2 y =7

−2x +4 y = −14⎧⎨⎪

⎩⎪

x =2 y +7

−2(2 y +7)+4 y = −14



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



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



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?



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


−2 y = −x +7



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


−2 y = −x +7

y =

12

x − 72



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


−2 y = −x +7

y =

12

x − 72

4 y =2x −14



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


−2 y = −x +7

y =

12

x − 72

4 y =2x −14

y =

12

x − 72



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


−2 y = −x +7

y =

12

x − 72

4 y =2x −14

y =

12

x − 72

These are the same lines!



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


−2 y = −x +7

y =

12

x − 72

4 y =2x −14

y =

12

x − 72


Infinitely many solutions on the line.



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


−2 y = −x +7

y =

12

x − 72

4 y =2x −14

y =

12

x − 72


Infinitely many solutions on the line.



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27



b. 2x −7 y = −2

−4x +14 y =3⎧⎨⎪

⎩⎪

−7 y = −2x −2

y =

27

x − 27

−4x +14 2

7x − 2

7⎛⎝⎜

⎞⎠⎟=3



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



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



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



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



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.



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.


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


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Int Math 2 Section 8-3 1011

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