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Solving Systems of Equations by Graphing

Any collection of two or more equations is called a system of equations. If the equations of a system involve two variables, then the set of ordered pairs that satisfy all of the equations is the solution set of the system. In this section we solve systems of linear equations in two variables.

Solving a System by Graphing

Because the graph of each linear equation is a line, points that satisfy both equations lie on both lines. For some systems these points can be found by graphing.

 

Example 1

A system with only one solution

Solve the system by graphing:

y = x + 2

x + y = 4

Solution

First write the equations in slope-intercept form:

y = x + 2

y = -x + 4

Use the y-intercept and the slope to graph each line. The graph of the system is shown in the figure below.

From the graph it appears that these lines intersect at (1, 3). To be certain, we can check that (1, 3) satisfies both equations. Let x = 1 and y = 3 in y = x + 2 to get 3 = 1 + 2.

Let x = 1 and y = 3 in x + y = 4 to get 1 + 3 = 4.

Because (1, 3) satisfies both equations, the solution set to the system is {(1, 3)}.

The graphs of the equations in the next example are parallel lines, and there is no point of intersection.

 

Example 2

A system with no solution

Solve the system by graphing:

2x - 3y = 6

3y - 2x = 3

Solution

First write each equation in slope-intercept form:

2x - 3y = 6 3y - 2x = 3
-3y = -2x + 6 3y = 2x + 3
y y

The graph of the system is shown in the next figure.

Because the two lines in this figure are parallel, there is no ordered pair that satisfies both equations. The solution set to the system is the empty set, Ø.

The equations in the next example are two equations that look different for the same straight line.

 

Example 3

A system with infinitely many solutions

Solve the system by graphing:

2(y + 2) = x

x - 2y = 4

Solution

Write each equation in slope-intercept form:

2(x + 2) = x x - 2y = 4
2y + 4 = x -2y = -x + 4
y y

Because the equations have the same slope-intercept form, the original equations are equivalent. Their graphs are the same straight line as shown in the following figure.

Every point on the line satisfies both equations of the system. There are infinitely many points in the solution set. The solution set is {(x, y) | x - 2y = 4}.

 
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