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# Solving Linear Systems of Equations by Graphing

One way to find the solution of a linear system in two variables is to graph each equation on the same coordinate axes.

If the lines intersect, the point(s) of intersection is the solution of the system.

Example 1

Graph each equation to find the solution of this system.

 3x - 2yy = 8= -x + 6

Solution

To graph each equation, first write it in slope-intercept form, y = mx + b.

 â€¢ Letâ€™s start with the first equation. Subtract 3x from both sides. 3x - 2y-2y = 8= -3x + 8 Divide both sides by -2. The y-intercept is (0, -4). Plot the point (0, -4). y The slope is This is the ratio To locate a second point, start at (0, -4), move up 3 (the rise) and then move right 2 (the run).

Plot the new point (2, -1). Finally, draw the line through (0, -4) and (2, -1).

Each point on this line represents a solution of 3x - 2y = 8. â€¢ The second equation, y = -x + 6, is given in the form y = mx + b.

The y-intercept is (0, 6). Plot the point (0, 6).

The slope is -1, which can be written as To locate a second point, start at (0, 6), move down 1 (the rise) and move right 1 (the run).

Plot the new point (1, 5).

Finally, draw the line through (0, 6) and (1, 5).

Every point on this line represents a solution of y = -x + 6. From the graph, it appears that the lines intersect at the point (4, 2).

The point (4, 2) is a solution of each equation.

Therefore, the solution of the system is (4, 2).

Letâ€™s verify that (4, 2) satisfies both equations.

 First equation Second equation Is Is Is 3x 3(4) 12 -- - 2y 2(2) 4 8 = 8= 8 ? = 8 ? = 8 ? Yes Is Is y (2) 2 == = -x + 6-(4) + 6 ? 2 ? Yes

Since (4, 2) satisfies both equations, it is the solution of the system.

The solution can be written as x = 4 and y = 2, or simply (4, 2).

A system that has at least one solution is called a consistent system.