EDGE Math 8

Solutions to Systems of Two Linear Equations in Two Variables

A solution to a system of two linear equations in two variables is an ordered pair of numbers satisfying both equations of the system.

Example

Determine if the ordered pair (1, 3) is a solution of the following system of two linear equations in two variables:

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

Solution

To determine if the ordered pair (1, 3) is a solution of the system, substitute 1 for x and 3 for y in each equation in the system. See if both equations are satisfied.

First equation:

$x + 2 y$	$= 7$
$1 + 2 (3)$	$= 7$
$7$	$= 7$	True

Second equation:

$- x + y$	$= 2$
$- 1 + 3$	$= 2$
$2$	$= 2$	True

The ordered pair (1, 3) satisfies both equations; hence, it is a solution of the system.

Recall that the graph of a linear equation in two variables is a line. Thus, the graphs of a system of two linear equations in two variables are two lines (one for each equation), which are graphed together on one set of axes. If two distinct lines intersect, the point at which they meet is called the point of intersection of the two lines. Such a point represents the solution of the system.