# Systems of Equations

## Elimination Using Addition and Subtraction

In systems of equations where the coefficients of terms containing the same variable are opposites, the elimination method can be applied by adding the equations. If the coefficients of those terms are the same, the elimination method can be applied by subtracting the equations.

Example

Solve the following system of equations using elimination.

x - 2y = 13 and 3x + 2y = 15

Solution

Add the two equations, since the coefficients of the y-terms, -2 and 2, are opposites.

 x - 2y = 13 (+) 3x + 2y = 15 4x = 28 Solve for x. x = 7 x - 2y = 13 Use the first equation. 7 - 2y = 13 Substitute 7 for x. - 2y = 6 y = -3

The solution of the system is (7, - 3).

## Elimination Using Multiplication

An extension of the elimination method is to multiply one or both of the equations in a system by some number so that adding or subtracting eliminates a variable.

Example

Solve the following system of equations using elimination.

x - y = 5 and 3x + 2y = 15

Solution

Multiply the first equation by 2 so that the coefficient of the y-terms in the system will be opposites. Then, add the equations and solve for x.

 2(x - y) = 2(5) 2x - 2y = 10 3x + 2y = 15 (+) 3x + 2y = 15 5x = 25 x = 5 x - y = 5 Use the first equation. 5 - y = 5 Substitute 5 for x. -y = 0 y = 0

The solution to this system is (5, 0).