11.2 Graphing linear equations

Recognize the relation between the solutions of an equation and its graph

In Use the Rectangular Coordinate System , we found a few solutions to the equation $3x+2y=6$ . They are listed in the table below. So, the ordered pairs $(0,3)$ , $(2,0)$ , $(1,\frac{3}{2})$ , $(4,-3)$ , are some solutions to the equation $3x+2y=6$ . We can plot these solutions in the rectangular coordinate system as shown on the graph at right.

Notice how the points line up perfectly? We connect the points with a straight line to get the graph of the equation $3x+2y=6$ . Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.

Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions!

Notice that the point whose coordinates are $(-2,6)$ is on the line shown in [link] . If you substitute $x=-2$ and $y=6$ into the equation, you find that it is a solution to the equation.

So $(4,1)$ is not a solution to the equation $3x+2y=6$ . Therefore the point $(4,1)$ is not on the line.

This is an example of the saying,” A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation $3x+2y=6$ .

The graph of $y=2x-3$ is shown below.

For each ordered pair decide

1. ⓐ Is the ordered pair a solution to the equation?
2. ⓑ Is the point on the line?

1. ⓐ $(0,3)$
2. ⓑ $(3,-3)$
3. ⓒ $(2,-3)$
4. ⓓ $(-1,-5)$

Substitute the $x$ - and $y$ -values into the equation to check if the ordered pair is a solution to the equation.

ⓐ

ⓑ Plot the points A: $(0,-3)$ B: $(3,3)$ C: $(2,-3)$ and D: $(-1,-5)$ .
The points $(0,-3)$ , $(3,3)$ , and $(-1,-5)$ are on the line $y=2x-3$ , and the point $(2,-3)$ is not on the line.

The points which are solutions to $y=2x-3$ are on the line, but the point which is not a solution is not on the line.

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The graph of $y=3x-1$ is shown.

For each ordered pair, decide

1. ⓐ is the ordered pair a solution to the equation?
2. ⓑ is the point on the line?

1. $(0,-1)$
2. $(2,2)$
3. $(3,-1)$
4. $(-1,-4)$

1. ⓐ yes ⓑ yes
2. ⓐ no ⓑ no
3. ⓐ no ⓑ no
4. ⓐ yes ⓑ yes

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Graph a linear equation by plotting points

There are several methods that can be used to graph a linear equation. The method we used at the start of this section to graph is called plotting points, or the Point-Plotting Method .

Let’s graph the equation $y=2x+1$ by plotting points.