# 11.2 Graphing linear equations

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By the end of this section, you will be able to:
• Recognize the relation between the solutions of an equation and its graph
• Graph a linear equation by plotting points
• Graph vertical and horizontal lines

Before you get started, take this readiness quiz.

1. Evaluate: $3x+2$ when $x=-1.$
If you missed this problem, review Multiply and Divide Integers .
2. Solve the formula: $5x+2y=20$ for $y.$
If you missed this problem, review Solve a Formula for a Specific Variable .
3. Simplify: $\frac{3}{8}\left(-24\right)\text{.}$
If you missed this problem, review Multiply and Divide Fractions .

## 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 $\left(0,3\right)$ , $\left(2,0\right)$ , $\left(1,\frac{3}{2}\right)$ , $\left(4,-3\right)$ , 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 $\left(-2,6\right)$ 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 $\left(4,1\right)$ is not a solution to the equation $3x+2y=6$ . Therefore the point $\left(4,1\right)$ 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 $3x+2y=6$ .

## Graph of a linear equation

The graph of a linear equation $Ax+By=C$ is a straight line.
• Every point on the line is a solution of the equation.
• Every solution of this equation is a point on this line.

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. $\left(0,3\right)$
2. $\left(3,-3\right)$
3. $\left(2,-3\right)$
4. $\left(-1,-5\right)$

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: $\left(0,-3\right)$ B: $\left(3,3\right)$ C: $\left(2,-3\right)$ and D: $\left(-1,-5\right)$ .
The points $\left(0,-3\right)$ , $\left(3,3\right)$ , and $\left(-1,-5\right)$ are on the line $y=2x-3$ , and the point $\left(2,-3\right)$ 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.

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. $\left(0,-1\right)$
2. $\left(2,2\right)$
3. $\left(3,-1\right)$
4. $\left(-1,-4\right)$
1. yes yes
2. no no
3. no no
4. yes yes

## 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.

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where no alternative is available
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