5-4 Elimination Using Multiplication

aka Linear Combination

Algebra 1 Glencoe McGraw-Hill Linda Stamper

12y4x413y8x3

Often the equations are not ready for one variable to cancel. You will need to create the opposites.

Multiply one or both equations by a number to obtain coefficients that are opposites for one of the variables.

12y4x413y8x3

Choose one of the variables to create an opposite.

Multiply by a number needed to make one variable opposites.

212y4x42 24y8x8

Add your equation because you now have opposites.

x11 111x

13y8x3

13y81 3

2y16y813y83

Write solution as an ordered pair.(–1,2)

Substitute the solved value into either of the originaloriginal equations to find the value for the other variable.

13y8x 3

Yeah a handout. I do not have to

copy these notes in my notebook!

1328x 3

12y4x4

13y8x3

413y8x34 52y32x12 36 y12x12

y44 882y

13y8x 3

1x

3x3

1616

1316x3

(-1,2)

312y4x43

While you can choose either of the variables to make opposites, choosing wisely may save you some work. Here is the work if the x variable is used to create the opposite.

8y3x78y x3

Example 1

Solve the linear system using elimination.

Example 2

19y5x26y4x3

8y3x78y x3

Example 1 Solve the linear system.

38y x33 24y3x9

8 y3x7

x16 322x

8yx 3

8y2 3

2y8y6

(2,2)

6y42 3

19y5x26y4x3

Example 2 Solve the linear system. 56y4x35 30y20x15

76 y20x8

x23 46 2x

6y4x 3

3y12y46y46

(–2,3)

419y5x24

634x 3

12y8x6 19y5x2

6y4x3

Example 2 Solve the linear system. 26y4x32

75 y15x6

y23 693y

6y4x 3

2x

6x3

612x3

(–2,3)

319y5x23

Solving A Linear System By Elimination

1) Arrange the equations with like terms in columns.

2) Multiply, if necessary, one or both equations by the number needed to make one of the variables an opposite.

3) Add the equations when one of the variables have opposites. Then solve.

4) Substitute the value solved into either of the original equations and solve for the other variable.

5) Check the ordered pair solution in each of the original equations.

6y3x

When solving a system by elimination, rearrange the terms so that the corresponding variables are vertically stacked.

6y3x12y6x3

12y6x3

)3(6y3x3 18y9x3

12y6x3

30y15 2y Substitute the solved value

into either of the original equations.

6y 3x

6)2(3x

0x0x66x

Write answer as an ordered

pair. (0,2)

Example 3

Solve the linear system using elimination.

Example 4

x512y28y2x3

1y2x5y3x413

Example 5

18y2x615x5y3

Example 6 Write a linear system and then solve.

Five times the first number minus three times the second number is six. Two times the first number minus five times the second number is ten. Find the numbers.Assign

labels.Translate each sentence.Solve the system.Write a sentence to give the answer.

x512y28y2x3

Example 3 Solve the linear system. 8y2x3

12y2x5

18y2x31 8y2x3

12y2x5

2x4x2

8y2x 3

8y2)2( 3

y1y22

8y26

(2,1)

Rewrite in standard form.

1y2x5y3x413

Example 4 Solve the linear system.

Rewrite in standard form.

13y3x4 1y2x5

26y6x8 3y6x15

1x23x23

y3x 413

y31413

y3y39

y3413

(–1,3)

18y2x615x5y3

18y26x 15y3x5

54y68x1

30y6x10

24x8 3x

0,3

Example 5 Solve the linear system. 15x5y3

0y30

y

0y3

1515y3

1535y3

50y25x10 12y6x10 6y3x5

Example 6 Five times the first number minus 3 times the second number is six. Two times the first number minus five times the second number is 10. Find the numbers.

y19 382y

Let x = first numberLet y = second numberTranslate each sentence.Solve the system.

62 3x5

10y5x2

Write a sentence to give the answer.

The numbers are -2 and 0.

6y3x5

66x5 0x5 0x

Practice Problems

8y5x44y3x2 )1

9y3x3

2y2x6 )2

2y8x2

3y4x5 )3

1) (2,0) 2) (1,–2) 3)

31

,31

4) Six times the first number plus two times the second number is two. Four times the first number plus three times the second number is eight. Find the numbers. The numbers are -1 and

4.

5-A6 Page 276-278 #7–17,30,34-36,44-48.