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Chapter 8.3. Solving a System of Equations in Two Variables By Elimination. Steps to solve a system of equations using the elimination method. The coefficients of one variable must be opposite. You may have to multiply one or both equations by an integer so that step 1 occurs .

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Solving a system of equations in two variables by elimination

Chapter 8.3

Solving a System of Equations in Two Variables By Elimination


Solving a system of equations in two variables by elimination

Steps to solve asystem of equations

using the elimination method

The coefficients of one variable must be opposite.

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

Add the equations so that a variable is eliminated.

Solve for the remaining variable.

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

Check the solution.


Solving a system of equations in two variables by elimination

1. Solve by addition.

3x

+ y

= 7

5x

– 2y

= 8

step 1 coefficients of one variable must be opposite.


Solving a system of equations in two variables by elimination

1. Solve by addition.

3x

+ y

2( )

= 7

5x

– 2y

= 8

6x + 2y = 14

5x – 2y = 8

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


Solving a system of equations in two variables by elimination

1. Solve by addition.

3x

+ y

2( )

= 7

5x

– 2y

= 8

6x + 2y = 14

5x – 2y = 8

11x = 22

step 3 add to eliminate the y.


Solving a system of equations in two variables by elimination

1. Solve by addition.

3x

+ y

2( )

= 7

5x

– 2y

= 8

6x + 2y = 14

5x – 2y = 8

11x = 22

11 11

x = 2

step 4 solve for x.


Solving a system of equations in two variables by elimination

1. Solve by addition.

3x

+ y

2( )

= 7

3(2)

+ y

= 7

5x

– 2y

= 8

6 + y = 7

-6 -6

6x + 2y = 14

5x – 2y = 8

y = 1

11x = 22

11 11

(2, 1)

x = 2

step 5 substitute into equation 1 and solve for y.


Solving a system of equations in two variables by elimination

2. Solve by addition.

4x

+ 5y

= 17

3x

+ 7y

= 12

step 1 coefficients of one variable must be opposite.


Solving a system of equations in two variables by elimination

2. Solve by addition.

4x

+ 5y

3( )

= 17

-4( )

3x

+ 7y

= 12

12x + 15y = 51

-12x – 28y = -48

step 2 make the x opposites, multiply first equation by 3,

second equation by -4.


Solving a system of equations in two variables by elimination

2. Solve by addition.

4x

+ 5y

3( )

= 17

-4( )

3x

+ 7y

= 12

12x + 15y = 51

-12x – 28y = -48

-13y = 3

step 3 add to eliminate the x.


Solving a system of equations in two variables by elimination

2. Solve by addition.

4x

+ 5y

3( )

= 17

-4( )

3x

+ 7y

= 12

12x + 15y = 51

-12x – 28y = -48

-13y = 3

-13 -13

-3

y =

13

step 4 solve for y.


Solving a system of equations in two variables by elimination

2. Solve by addition.

4x

+ 5y

3( )

= 17

4x + ( ) = 17

4x

= 17

+ 5( )

-4( )

3x

+ 7y

= 12

13( )

52x

– 15

= 221

12x + 15y = 51

+15 +15

-12x – 28y = -48

52x = 236

-13y = 3

52 52

-13 -13

59

-3

x =

y =

( , )

13

13

-3

-15

13

13

step 5 substitute into equation 1 and solve for x.

-3

59

13

13


Solving a system of equations in two variables by elimination

3. Solve by addition.

x

= 3

– y

12

( )

-2x

+ y

= 6

8x

– 9y

= 36

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

step 1 coefficients of one variable must be opposite.


Solving a system of equations in two variables by elimination

3. Solve by addition.

x

= 3

– y

12( )

-2x

+ y

= 6

4( )

8x

– 9y

= 36

-8x

+ 4y

= 24

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


Solving a system of equations in two variables by elimination

3. Solve by addition.

x

= 3

– y

12( )

-2x

+ y

= 6

4( )

8x

– 9y

= 36

-8x

+ 4y

= 24

-5y = 60

step 3 add to eliminate the x.


Solving a system of equations in two variables by elimination

3. Solve by addition.

x

= 3

– y

12( )

-2x

+ y

= 6

4( )

8x

– 9y

= 36

-8x

+ 4y

= 24

-5y = 60

-5 -5

y = -12

step 4 solve for y.


Solving a system of equations in two variables by elimination

3. Solve by addition.

-2x

x

+ (-12)

= 6

= 3

– y

12( )

+12 +12

-2x

+ y

= 6

4( )

-2x = 18

8x

– 9y

= 36

-2 -2

-8x

+ 4y

= 24

x = -9

-5y = 60

-5 -5

(-9, -12)

y = -12

step 5 substitute into equation 2 and solve for x.


Solving a system of equations in two variables by elimination

4. Solve by addition.

0.2x

= -0.1

+ 0.3y

10

( )

( )

10

0.5x

– 0.1y

= -1.1

2x

+3y

= -1

5x

– y

= -11

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

step 1 coefficients of one variable must be opposite.


Solving a system of equations in two variables by elimination

4. Solve by addition.

2x

+3y

= -1

3( )

5x

– y

= -11

2x + 3y = -1

15x

– 3y

= -33

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


Solving a system of equations in two variables by elimination

4. Solve by addition.

2x

+3y

= -1

3( )

5x

– y

= -11

2x + 3y = -1

15x

– 3y

= -33

17x = -34

step 3 add to eliminate the y.


Solving a system of equations in two variables by elimination

4. Solve by addition.

2x

+3y

= -1

3( )

5x

– y

= -11

2x + 3y = -1

15x

– 3y

= -33

17x = -34

17 17

x = -2

step 4 solve for x.


Solving a system of equations in two variables by elimination

4. Solve by addition.

2(-2)

2x

+ 3y

= -1

+3y

= -1

3( )

5x

– y

= -11

-4 + 3y = -1

+4 +4

2x + 3y = -1

3y = 3

15x

– 3y

= -33

3 3

17x = -34

17 17

y = 1

(-2, 1)

x = -2

step 5 substitute into equation 1 and solve for y.


Solving a system of equations in two variables by elimination

Chapter 8.3

Solving a System of Equations in Two Variables By Elimination