Solving a System of Equations in Two Variables By Elimination

Solving a System of Equations in Two Variables By Elimination

Chapter 8.3

Steps to solve a system of equationsusing the elimination method

1. The coefficients of one variable must be opposite.

2. You may have to multiply one or both equations by an integer so that step 1 occurs .

3. Add the equations so that a variable is eliminated.

4. Solve for the remaining variable.

5. Substitute the value into one of the original equations to solve for the other variable.

6. Check the solution.

3x5x

step 1 coefficients of one variable must be opposite.

1. Solve by addition.

+ y = 7– 2y = 8

2( )

6x + 2y = 145x – 2y = 8

step 2 make the y opposites, multiply first equation by 2.

1. Solve by addition.

3x5x

+ y = 7– 2y = 8

2( )

6x + 2y = 145x – 2y = 8

11x = 22

step 3 add to eliminate the y.

1. Solve by addition.

3x5x

+ y = 7– 2y = 8

2( )

6x + 2y = 145x – 2y = 8

11x = 2211 11

x = 2

step 4 solve for x.

1. Solve by addition.

3x5x

+ y = 7– 2y = 8

2( )

6x + 2y = 145x – 2y = 8

11x = 2211 11

x = 2

3(2)

-6 -6

(2, 1)

6 + y = 7

y = 1

step 5 substitute into equation 1 and solve for y.

1. Solve by addition.

+ y = 73x5x

+ y = 7– 2y = 8

4x3x

step 1 coefficients of one variable must be opposite.

2. Solve by addition.

+ 5y = 17+ 7y = 12

-4( )3( )

12x + 15y = 51-12x – 28y = -48

step 2 make the x opposites, multiply first equation by 3, second equation by -4.

2. Solve by addition.

4x3x

+ 5y = 17+ 7y = 12

-4( )

-13y = 3

step 3 add to eliminate the x.

2. Solve by addition.

3( )

12x + 15y = 51-12x – 28y = -48

4x3x

+ 5y = 17+ 7y = 12

-4( )

-13 -13

step 4 solve for y.

2. Solve by addition.

-13y = 3

3( )

12x + 15y = 51-12x – 28y = -48

y = 13-3

4x3x

+ 5y = 17+ 7y = 12

-4( )

52x

13( )

step 5 substitute into equation 1 and solve for x.

2. Solve by addition.

-13 -13-13y = 3

3( )

12x + 15y = 51-12x – 28y = -48

y = 13-3

4x + ( ) = 1713-15

+15 +15

52x = 23652 52

x = 1359( , )

135913

-3

– 15 = 221

+ 5( )13-3 = 174x4x

3x+ 5y = 17+ 7y = 12

( )12 x-2x

Before beginning with the steps remove the fractions in the first equation by multiplying 12 to each term.

3. Solve by addition.

8x – 9y = 36

step 1 coefficients of one variable must be opposite.

– y = 3+ y = 6

4( )12( )

3. Solve by addition.

8x – 9y = 36

step 2 make x opposites, multiply second equation by 4.

-8x + 4y = 24

x-2x

– y = 3+ y = 6

4( )

3. Solve by addition.

12( )

8x – 9y = 36-8x + 4y = 24

-5y = 60

step 3 add to eliminate the x.

x-2x

– y = 3+ y = 6

4( )12( )

3. Solve by addition.

8x – 9y = 36-8x + 4y = 24

-5y = 60-5 -5

step 4 solve for y.

y = -12

x-2x

– y = 3+ y = 6

4( )12( )

3. Solve by addition.

8x – 9y = 36-8x + 4y = 24

-5y = 60-5 -5

step 5 substitute into equation 2 and solve for x.

y = -12

-2x+12 +12

-2x = 18-2 -2

+ (-12) = 6

x = -9

(-9, -12)

x-2x

– y = 3+ y = 6

( )( )10

100.2x0.5x

Before beginning with the steps remove the decimals by multiplying 10 to each term in each equation.

4. Solve by addition.

2x + 3y = -1

step 1 coefficients of one variable must be opposite.

5x – y = -11

+ 0.3y = -0.1– 0.1y = -1.1

3( )

4. Solve by addition.

2x + 3y = -1

step 2 make y opposites, multiply second equation by 3.

5x – y = -11

2x + 3y = -115x – 3y = -33

3( )

4. Solve by addition.

2x + 3y = -15x – y = -11

2x + 3y = -115x – 3y = -3317x = -34

step 3 add to eliminate the y.

3( )

4. Solve by addition.

2x + 3y = -15x – y = -11

2x + 3y = -115x – 3y = -3317x = -3417 17

step 4 solve for x.

x = -2

3( )

4. Solve by addition.

2x + 3y = -15x – y = -11

2x + 3y = -115x – 3y = -3317x = -3417 17

x = -2

step 5 substitute into equation 1 and solve for y.

2(-2)

+4 +43y = 33 3

+ 3y = -1

y = 1

(-2, 1)

-4 + 3y = -1

Solving a System of Equations in Two Variables By Elimination

Chapter 8.3