Systems By Elimination PPT.ppt

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Solving Systems of

Equations

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Objectives

• Learn the procedure of the Elimination

Method using addition

• Learn the procedure of the Elimination

Method using multiplication

• Solving systems of equations using the

Elimination Method

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When to use

Elimination Method

• Elimination works because opposites

cancel each other out

– Example 3x + (-3x) = 0

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Can you eliminate?

Consider the system x - 2y = 5

2x + 2y = 7

REMEMBER: We are trying to find the Point of Intersection. (x, y)

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Step 1 – Eliminate like terms

2x + 2y = 7

Lets add both equations to each other

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Step 2 – Cancel & solve

2x + 2y = 7

Lets add both equations to each other

+

3x = 12 x = 4 

ANS: (4, y)

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Step 3 – Substitute & solve

the other coefficient

Consider the system

2x + 2y = 7

Lets substitute x = 4 into this equation.

4 - 2y = 5 Solve for y - 2y = 1

y = 1

2 

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Which equation should we

substitute with?

ANS: (4, )

We could have substituted x = 4

into this equation (but it is harder) 2(4) + 2y = 7 Solve for y

8 + 2y = 7 -8 -8 2y = -1 2 2



1 2

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Step 4 - Check

? ?

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Example 2

Consider the system 3x + y = 14

4x - y = 7

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Example 2

Consider the system 3x + y = 14

4x - y = 7 7x = 21

x = 3 

ANS: (3, y)

+

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Example 2

Consider the system

ANS: (3, )

3x + y = 14 4x - y = 7

Substitute x = 3 into this equation

3(3) + y = 14 9 + y = 14

y = 5

5



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Example 2

Step 4: Check

Is the solution (3,5) ? Substitute & solve to check

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Your Turn…

2x y

+

=

5 3x y



₌

₁₅

1. 2.

2y x



=

5

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Your Turn (Answers)

2x y

+

=

5 3x y



₌

₁₅

1. 2.

2y x



=

5 6y x

+

=

11

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Elimination using

Multiplication

Consider the system

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Example 3) Elimination using

Multiplication

Consider the system

6x + 11y = -5 6x + 9y = -3

+

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Example 3) Elimination using

Multiplication

Consider the system

6x + 11y = -5

6x + 9y = -3 _{Wouldn’t it be}

easy to make the x coefficients

opposites?

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Example 3) Elimination using

Multiplication

Consider the system

6x + 11y = -5 6x + 9y = -3

-1 ( ) _Distribute

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Example 3) Elimination using

Multiplication

Consider the system

- 6x - 11y = 5 6x + 9y = -3

+

-2y = 2 y = -1

ANS: (x, )-1 

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Example 3

Consider the system

6x + 11y = -5 6x + 9y = -3

ANS: (x, )-1

y = -1

Lets substitute y = -1 into this equation

6x + 9(-1) = -3 6x + -9 = -3

+9 +9 6x = 6

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Example 3

Consider the system

6x + 11y = -5 6x + 9y = -3

ANS: ( , )_-1

y = -1

Lets substitute y = -1 into this equation

6x + 9(-1) = -3 6x + -9 = -3

+9 +9 6x = 6

x = 1

1



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Example 3

Is the solution (1,-1)? Substitute & solve to check

Step 5: Check

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Example 4

Consider the system

x + 2y = 6 3x + 3y = -6

Multiply by -3 to eliminate the x term

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Example 4

Consider the system

x + 2y = 6 3x + 3y = -6

-3 ( ) _Distribute

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Example 4

Consider the system

-3x + -6y = -18

3x + 3y = -6

+

-3y = -24 y = 8

ANS: (x, 8) 

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Example 4

Consider the system

x + 2y = 6 3x + 3y = -6

Substitute y =14 into equation

y =8

x + 2(8) = 6 x + 16 = 6

x = -10 

Step 4: Substitute your answer to solve for the other coefficient

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Example 4

Is the solution (-10,8)? Substitute & solve to check

Step 5: Check

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Your Turn #2

1.

x + 2y = 5

2x + 6y = 12

2.

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Your Turn #2

1.

x + 2y = 5

2x + 6y = 12

2.

ANS: (3, 1)

x + 2y = 4 x - 4y = 16

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Example 5

Consider the system 3x + 4y = -25

2x - 3y = 6

2( ) -3( )

Notice – both

equations need to be multiplied to get opposite

coefficients.

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Example 5

Consider the system

6x + 8y = -50

-6x + 9y = -18

+

17y = -68 y = -4

ANS: (x, -4)



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Example 5

Consider the system

3x + 4y = -25 2x - 3y = 6

ANS: (-3, -4) Substitute y = -4

2x - 3(-4) = 6 2x - -12 = 6

2x + 12 = 6 2x = -6

x = -3 

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Example 5

Is the solution (-3,-4)? Substitute & solve to check

? ?

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Your Turn #3

1. 2.

4x + y = 9 3x + 2y = 8

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Your Turn #3

1. 2.

4x + y = 9 3x + 2y = 8

2x + 3y = 1 5x + 7y = 3