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# Solving a System of Equations in Two Variables By Elimination - PowerPoint PPT Presentation

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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Chapter 8.3

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.

3x

+ y

= 7

5x

– 2y

= 8

step 1 coefficients of one variable must be opposite.

3x

+ y

2( )

= 7

5x

– 2y

= 8

6x + 2y = 14

5x – 2y = 8

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

3x

+ y

2( )

= 7

5x

– 2y

= 8

6x + 2y = 14

5x – 2y = 8

11x = 22

step 3 add to eliminate the y.

3x

+ y

2( )

= 7

5x

– 2y

= 8

6x + 2y = 14

5x – 2y = 8

11x = 22

11 11

x = 2

step 4 solve for x.

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.

4x

+ 5y

= 17

3x

+ 7y

= 12

step 1 coefficients of one variable must be opposite.

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.

4x

+ 5y

3( )

= 17

-4( )

3x

+ 7y

= 12

12x + 15y = 51

-12x – 28y = -48

-13y = 3

step 3 add to eliminate the x.

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.

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

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.

x

= 3

– y

12( )

-2x

+ y

= 6

4( )

8x

– 9y

= 36

-8x

+ 4y

= 24

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

x

= 3

– y

12( )

-2x

+ y

= 6

4( )

8x

– 9y

= 36

-8x

+ 4y

= 24

-5y = 60

step 3 add to eliminate the x.

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.

-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.

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.

2x

+3y

= -1

3( )

5x

– y

= -11

2x + 3y = -1

15x

– 3y

= -33

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

2x

+3y

= -1

3( )

5x

– y

= -11

2x + 3y = -1

15x

– 3y

= -33

17x = -34

step 3 add to eliminate the y.

2x

+3y

= -1

3( )

5x

– y

= -11

2x + 3y = -1

15x

– 3y

= -33

17x = -34

17 17

x = -2

step 4 solve for x.

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.

Chapter 8.3

Solving a System of Equations in Two Variables By Elimination