EDGE Math 8

Elimination Method

To solve a system of two linear equations in two variables using the elimination method (also called the addition method), eliminate one of the variables in the system by producing a set of two new equations that are equivalent to those of the given system.

Example 1

Solve the system of linear equations using the elimination method. Determine what type of system of equations it is.

{\begin{array}{l} 2 x + 3 y = 7 \\ - 3 x - 2 y = 2 \end{array}

Solution

Eliminate one of the variables; in this case, eliminate x. To do this, multiply both sides of the first equation by 3 and the second equation by 2.

\begin{array}{l} 6 x + 9 y = 21 \\ - 6 x - 4 y = 4 \end{array}

The coefficients of x in the two new equations (which are 6 and −6) are additive inverses. If you add the two new equations in column form, the terms 6x and −6x will cancel out, thus resulting in an equation with only one variable y.

\begin{array}{l} 6 x + 9 y = 21 \\ \underline{+ - 6 x - 4 y = 4} \\ 5 y = 25 \end{array}

Then solve for y.

$5 y$	$= 25$
$y$	$= \frac{25}{5}$
$y$	$= 5$

To solve for x, substitute 5 for y in either the first or the second equation of the given system. Using the first equation and substituting 5 for y, you will get the following:

$2 x + 3 (5)$	$= 7$
$2 x + 15$	$= 7$
$2 x$	$= 7 - 15$
$2 x$	$= - 8$
$x$	$= \frac{- 8}{2}$
$x$	$= - 4.$

Hence, $x = - 4$ and $y = 5.$ Check the answer as an exercise. The solution is $(- 4, 5) .$ The system is consistent and independent.