5-3 Elimination Using Addition and Subtraction

Algebra 1 Glencoe McGraw-Hill Linda Stamper

When solving a linear system by substitution, you needed to isolate one of the variables.

Sometimes it is not easy to isolate one of the variables. What is the result when you try to isolate one of the variables in the system below?

7y3x4

13y2x5

When it is not easy to isolate one of the variables using the substitution method, it may be easier to solve the system by elimination.

When using this method, you will be adding the two equations together in order to eliminate one of the variables.

18y2 3

Elimination Using Addition

14y2x518y2x3

Add the equations if one of the

variables has opposites. Then solve. x8 32

4x Substitute the solved value into either of the original equations to find the value for the other variable.

18y2x3

3y6y218y212

Write the solution as an ordered pair.

(4,3)

Check by substitution in both original equations.

4

16y5x3

Use elimination to solve the system of equations.Example

1Example 2

Example 3

5y4x37y2x3

y6 122y

7y2x3

72 2x 3

1x3x374x3

(1,2)

19y3x5

4y2x5

y5 153y

4y2x5

43 2x 5

2x10x546x5

(–2,3)

31y5x2

16y5x3

x5 153x

16y53 3

5y

5y

25y5

16y59

(3,5)

When the coefficients of terms are the same, you can eliminate the terms by subtracting the equations. When using elimination with subtraction to solve systems of equations, do not forget to distribute the negative sign over the entire equation that is subtracted. Since subtraction is the same as adding the inverse, you should change the signs of the terms and then add to eliminate the variable.

6y2x5

Elimination Using Subtraction

22y2x9

6y2x5

Subtract the equations if

one of the variables have the same coefficients. Then solve.

x4 164x Substitute the solved value into

either of the original equations to find the value for the other variable.Write the solution as an ordered pair.

Check by substitution in both original equations.

-- -- --

6 y24 5

7y

14y2

6y220

(4,-7)

8yx6

14yx12

14yx12

Elimination Using Subtraction

x6 61x

-- ++ ++

14 y1 21

2y

2y

14y12

(-1,2)

Distribute the negative sign over the entire equation that is subtracted.

It will not matter which equation you

create the opposite!

18y4x3

28y4x2

28y4x2

7c7b8

11c3b8

Use elimination to solve the system of equations.Example

4Example 5

c4 41c

11c3b8

111 3b8

x5 102x

28y42 2

6y

24y4

28y44

(2,6)

-- -- --

1,

4

7

++ -- --

113b8 14b8

814

b

47

b

18yx2 18yx2

Write and Solve a System of Equations

Twice one number added to another number is 18. Four times the first number minus the other number is 12. Find the numbers.

x6 305x

Assign Labels.

Translate each sentence.Solve the system. 18 y5 2

8y

18y10

12yx4

Write a sentence to give the answer.

The numbers are 5 and 8.

Let x = first numberLet y = second number

Example 6 The sum of two numbers is -10. Negative three times the first number minus the second number equals 2.Find the numbers.

Assign labels.

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

Example 7 Four times one number minus three times another number is 12. Two times the first number added to three times the second number is 6. Find the numbers.

10yx 10yx

Example 6 The sum of two numbers is -10. Negative three times the first number minus the second number equals 2.Find the numbers.

x2 84x

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

10 y4 14y

2yx3

Write a sentence to give the answer.The numbers are 4 and -14.

12y3x4 12y3x4

Example 7 Four times one number minus three times another number is 12. Two times the first number added to three times the second number is 6. Find the numbers.

x6 183x

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

12 y334 12y312

6y3x2

Write a sentence to give the answer.

The numbers are 3 and 0.

0y3 0y

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