# 4.2 Graph linear equations in two variables

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By the end of this section, you will be able to:
• Recognize the relationship 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 [link] .
2. Solve $3x+2y=12$ for $y$ in general.
If you missed this problem, review [link] .

## Recognize the relationship between the solutions of an equation and its graph

In the previous section, we found several solutions to the equation $3x+2y=6$ . They are listed in [link] . So, the ordered pairs $\left(0,3\right)$ , $\left(2,0\right)$ , and $\left(1,\frac{3}{2}\right)$ are some solutions to the equation $3x+2y=6$ . We can plot these solutions in the rectangular coordinate system as shown in [link] .

 $3x+2y=6$ $x$ $y$ $\left(x,y\right)$ 0 3 $\left(0,3\right)$ 2 0 $\left(2,0\right)$ 1 $\frac{3}{2}$ $\left(1,\frac{3}{2}\right)$

Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation $3x+2y=6$ . See [link] . 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 the point $\left(-2,6\right)$ is a solution to the equation $3x+2y=6$ . (The phrase “the point whose coordinates are $\left(-2,6\right)$ ” is often shortened to “the point $\left(-2,6\right)$ .”)

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. See [link] . 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 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.

For each ordered pair, decide:

Is the ordered pair a solution to the equation?
Is the point on the line?

A $\left(0,-3\right)$  B $\left(3,3\right)$  C $\left(2,-3\right)$  D $\left(-1,-5\right)$

## Solution

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

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

Use the graph of $y=3x-1$ to decide whether each ordered pair is:

• a solution to the equation.
• on the line.

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

yes, yes  yes, yes

Use graph of $y=3x-1$ to decide whether each ordered pair is:

• a solution to the equation
• on the line

$\left(3,-1\right)$ $\left(-1,-4\right)$

no, no  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 to graph $3x+2y=6$ is called plotting points, or the Point–Plotting Method.

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