Solve the following system using the elimination method.

5x + y = 11 6x + 2y = 14

Notice that the x and y terms do not contain opposites.

6x + 2y = 14

If the 1st equation is multiplied by – 2, the y terms will become opposites.

5x + y = 11 Solve for y.

– 4x = – 8

→ 10 + y = 11

Solution: x = 2 and y = 1

→ y = 1

Add the two equations and solve.

5(2) + y = 11

x = 2

– 10x – 2y = – 22

10.05

Solving Systems of Equations

by Elimination

(2 Multipliers)

Remember that to solve a system of equations using the elimination method, it must have at least one set of opposite variable terms.

To solve this type of system, multiply the coefficients of “x” together and find this number.

Some systems require that both equations be changed in order to create opposites.

Then multiply the 1st by some factor that results in a positive value of that number. Multiply the 2nd equation by some factor that results in a negative value of that number.

Solve as usual.

Solve the following system using the elimination method.

4x + 5y = 3 5x + 2y = 8

Notice that the x and y terms do not contain opposites.

20x + 25y = 15

Both equations need to be changed in order to create opposites.

17y = – 17Solve for x.

4x + 5(– 1) = 3 → 4x – 5 = 3

→ 4x = 8

Solution: x = 2 and y = – 1

→ x = 2

Add the two equations and solve.

4x + 5y = 3

→ y = – 1

– 20x – 8y = – 32

Multiply the 1st equation by 5.Multiply the 2nd equation by – 4.

Solve the following system using the elimination method.

3x + 5y = 22 4x – 2y = 12

Notice that the x and y terms do not contain opposites.

12x + 20y = 88

Both equations need to be changed in order to create opposites.

26y = 52Solve for x.

3x + 5(2) = 22 → 3x + 10 = 22

→ 3x = 12

Solution: x = 4 and y = 2

→ x = 4

Add the two equations and solve.

3x + 5y = 22

→ y = 2

– 12x + 6y = – 36

Multiply the 1st equation by 4.Multiply the 2nd equation by – 3.

Solve the following system using the elimination method.

2x + 4y = 6 3x + 6y = 9

Notice that the x and y terms do not contain opposites.

6x + 12y = 18

Both equations need to be changed in order to create opposites.

0 = 0

There are infinite solutions to the system.

Add the two equations and solve.

The variables have cancelled and the statement is true.

– 6x – 12y = – 18

Multiply the 1st equation by 3.Multiply the 2nd equation by – 2.

Solve the following system using the elimination method.

– 2x – 2y = 4 3x + 3y = – 7

Notice that the x and y terms do not contain opposites.

6x + 6y = – 12

Both equations need to be changed in order to create opposites.

0 = 2

There are no solutions to the system.

Add the two equations and solve.

The variables have cancelled and the statement is false.

– 6x – 6y = 14

Multiply the 1st equation by – 3.Multiply the 2nd equation by – 2.

Try This:Solve the following system using the elimination method.

2x – 3y = 75x + 2y = – 11

Notice that the x and y terms do not contain opposites.

10x – 15y = 35

Both equations need to be changed in order to create opposites.

– 19y = 57

Solve for x.

2x – 3(– 3) = 7 → 2x + 9 = 7→ 2x = – 2

Solution: x = – 1 and y = – 3

→ x = – 1

Add the two equations and solve.

2x – 3y = 7

→ y = – 3

– 10x – 4y = 22Multiply the 1st equation by 5.Multiply the 2nd equation by – 2.