3.2 Graph Linear Equations in Two Variables

Izabela Mazur; Kim Moshenko

3. Equations and their Graphs

3.2 Graph Linear Equations in Two Variables

Learning Objectives

By the end of this section it is expected that 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.

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 the table below. So, the ordered pairs $\left(0,3\right)$ , $\left(2,0\right)$ , and $\left(1,\dfrac{3}{2}\right)$ are some solutions to the equation $3x+2y=6$ . We can plot these solutions in the rectangular coordinate system as shown in (Figure 1).

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

A graph that plots the points (0, 3), (1, three halves), and (2, 0). — Figure .1

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 (Figure 2). Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.

Described in previous paragraph. — Figure .2

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 (Figure 3). If you substitute $x=-2$ and $y=6$ into the equation, you find that it is a solution to the equation.

Graphs the equation 3x plus 2y equals 6. The points (negative 2, 6) and (4, 1) are plotted. The line goes through (−2, 6) but not (4, 1). — Figure .3

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 (Figure 2). 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.

EXAMPLE 1

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

Graphs the line 2x−3.

For each ordered pair, decide:

a) Is the ordered pair a solution to the equation?
b) 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.

a)

b) Plot the points A $\left(0,3\right)$ , B $\left(3,3\right)$ , C $\left(2,-3\right)$ , and D $\left(-1,-5\right)$ .
Graph of the equation 2x−3. The points described in the previous paragraph are plotted.

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.

TRY IT 1

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

a solution to the equation.
on the line.

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

Graph of the equation y = 3x−1.

Show answer

a) yes, yes b) 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.

EXAMPLE 2

How To Graph an Equation By Plotting Points

Graph the equation $y=2x+1$ by plotting points.

Solution

TRY IT 2

Graph the equation by plotting points: $y=2x-3$ .

Show answer

HOW TO: Graph a linear equation by plotting points.

The steps to take when graphing a linear equation by plotting points are summarized below.

Find three points whose coordinates are solutions to the equation. Organize them in a table.
Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.

If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between part (a) and part (b) in (Figure 4).

Figure a shows three points with a straight line through them. Figure b shows three points that do not lie on the same line. — Figure .4

Let’s do another example. This time, we’ll show the last two steps all on one grid.

EXAMPLE 3

Graph the equation $y=-3x$ .

Solution

Find three points that are solutions to the equation. Here, again, it’s easier to choose values for $x$ . Do you see why?

We list the points in the table below.

$y=-3x$
$x$	$y$	$\left(x,y\right)$
0	0	$\left(0,0\right)$
1	$-3$	$\left(1,-3\right)$
$-2$	6	$\left(-2,6\right)$

Plot the points, check that they line up, and draw the line.

Graph of the equation y = −3x. The points listed in the previous table are plotted.

TRY IT 3

Graph the equation by plotting points: $y=-4x$ .

Show answer

When an equation includes a fraction as the coefficient of $x$ , we can still substitute any numbers for $x$ . But the math is easier if we make ‘good’ choices for the values of $x$ . This way we will avoid fraction answers, which are hard to graph precisely.

EXAMPLE 4

Graph the equation $y=\dfrac{1}{2}x+3$ .

Solution

Find three points that are solutions to the equation. Since this equation has the fraction $\dfrac{1}{2}$ as a coefficient of $x,$ we will choose values of $x$ carefully. We will use zero as one choice and multiples of 2 for the other choices. Why are multiples of 2 a good choice for values of $x$ ?

The points are shown in the table below.

**$y=\dfrac{1}{2}x+3$**
$x$	$y$	$\left(x,y\right)$
0	3	$\left(0,3\right)$
2	4	$\left(2,4\right)$
4	5	$\left(4,5\right)$

Plot the points, check that they line up, and draw the line.

The points listed in the previous table are plotted. The equation y = 1 half x + 3 is graphed.

TRY IT 4

Graph the equation $y=\dfrac{1}{3}x-1$ .

Show answer

A graph of the equation y = 1 third x−1.

So far, all the equations we graphed had $y$ given in terms of $x$ . Now we’ll graph an equation with $x$ and $y$ on the same side. Let’s see what happens in the equation $2x+y=3$ . If $y=0$ what is the value of $x$ ?

This point has a fraction for the x– coordinate and, while we could graph this point, it is hard to be precise graphing fractions. Remember in the example $y=\dfrac{1}{2}x+3$ , we carefully chose values for $x$ so as not to graph fractions at all. If we solve the equation $2x+y=3$ for $y$ , it will be easier to find three solutions to the equation.

$\begin{array}{ccc}\hfill 2x+y& =\hfill & 3\hfill \\ \hfill y& =\hfill & -2x+3\hfill \end{array}$

The solutions for $x=0$ , $x=1$ , and $x=-1$ are shown in the table below. The graph is shown in (Figure 5).

**$2x+y=3$**
$x$	$y$	$\left(x,y\right)$
0	3	$\left(0,3\right)$
1	1	$\left(1,1\right)$
$-1$	5	$\left(-1,5\right)$

The points listed in the previous table are plotted. The equation 2x + y = 3 is graphed. — Figure .5

Can you locate the point $\left(\dfrac{3}{2},0\right)$ , which we found by letting $y=0$ , on the line?

EXAMPLE 5

Graph the equation $3x+y=-1$ .

Solution

Find three points that are solutions to the equation.	$3x+y\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-1$
First, solve the equation for $y$ .	$y\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-3x-1$

We’ll let $x$ be 0, 1, and $-1$ to find 3 points. The ordered pairs are shown in the table below. Plot the points, check that they line up, and draw the line. See (Figure 6).

**$3x+y=-1$**
$x$	$y$	$\left(x,y\right)$
0	$-1$	$\left(0,-1\right)$
1	$-4$	$\left(1,-4\right)$
$-1$	2	$\left(-1,2\right)$

The points listed in the previous table are plotted. The equation 3x+y = −1 is graphed. — Figure .6

EXAMPLE 5

Graph the equation $2x+y=2$ .

Show answer

If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the x– and y-axis are the same, the graphs match!

The equation in (Example 5) was written in standard form, with both $x$ and $y$ on the same side. We solved that equation for $y$ in just one step. But for other equations in standard form it is not that easy to solve for $y$ , so we will leave them in standard form. We can still find a first point to plot by letting $x=0$ and solving for $y$ . We can plot a second point by letting $y=0$ and then solving for $x$ . Then we will plot a third point by using some other value for $x$ or $y$ .

EXAMPLE 6

Graph the equation $2x-3y=6$ .

Solution

Find three points that are solutions to the equation.	$\begin{array}{ccc}\hfill 2x-3y& =\hfill & 6\hfill \end{array}$
First, let $x=0$ .	$\begin{array}{ccc}\hfill 2\left(0\right)-3y& =\hfill & 6\hfill \end{array}$
Solve for $y$ .	$\begin{array}{ccc}\hfill -3y& =\hfill & 6\hfill \\ \hfill y& =\hfill & -2\hfill \end{array}$
Now let $y=0$ .	$\begin{array}{ccc}\hfill 2x-3\left(0\right)& =\hfill & 6\hfill \end{array}$
Solve for $x$ .	$\begin{array}{ccc}\hfill 2x& =\hfill & 6\hfill \\ \hfill x& =\hfill & 3\hfill \end{array}$
We need a third point. Remember, we can choose any value for $x$ or $y$ . We’ll let $x=6$ .	$\begin{array}{ccc}\hfill 2\left(6\right)-3y& =\hfill & 6\hfill \end{array}$
Solve for $y$ .	$\begin{array}{ccc}\hfill 12-3y& =\hfill & 6\hfill \\ \hfill -3y& =\hfill & -6\hfill \\ \hfill y& =\hfill & 2\hfill \end{array}$

We list the ordered pairs in the table below. Plot the points, check that they line up, and draw the line. See (Figure 7).

**$2x-3y=6$**
$x$	$y$	$\left(x,y\right)$
0	$-2$	$\left(0,-2\right)$
3	0	$\left(3,0\right)$
6	2	$\left(6,2\right)$

The points listed in previous table are plotted. The equation 2x − 3y = 6 is plotted. — Figure .7

TRY IT 6

Graph the equation $4x+2y=8$ .

Show answer

Graph Vertical and Horizontal Lines

Can we graph an equation with only one variable? Just $x$ and no $y$ , or just $y$ without an $x$ ? How will we make a table of values to get the points to plot?

Let’s consider the equation $x=-3$ . This equation has only one variable, $x$ . The equation says that $x$ is always equal to $-3$ , so its value does not depend on $y$ . No matter what $y$ is, the value of $x$ is always $-3$ .

So to make a table of values, write $-3$ in for all the $x$ values. Then choose any values for $y$ . Since $x$ does not depend on $y$ , you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y-coordinates. See the table below.

**$x=-3$**
$x$	$y$	$\left(x,y\right)$
$-3$	1	$\left(-3,1\right)$
$-3$	2	$\left(-3,2\right)$
$-3$	3	$\left(-3,3\right)$

Plot the points from the table and connect them with a straight line. Notice in (Figure 8) that we have graphed a vertical line.

The points listed in the previous table are plotted. The equation x = −3 is graphed. The resulting line is vertical. — Figure .8

Vertical line

A vertical line is the graph of an equation of the form $x=a$ .

The line passes through the x-axis at $\left(a,0\right)$ .

EXAMPLE 7

Graph the equation $x=2$ .

Solution

The equation has only one variable, $x$ , and $x$ is always equal to 2. We create the table below where $x$ is always 2 and then put in any values for $y$ . The graph is a vertical line passing through the x-axis at 2. See (Figure 9).

**$x=2$**
$x$	$y$	$\left(x,y\right)$
2	1	$\left(2,1\right)$
2	2	$\left(2,2\right)$
2	3	$\left(2,3\right)$

The points listed in the previous table are plotted. The equation x = 2 is graphed. The resulting line is vertical. — Figure .9

TRY IT 7

Graph the equation $x=5$ .

Show answer

Graph of the equation x = 5. The resulting line is vertical.

What if the equation has $y$ but no $x$ ? Let’s graph the equation $y=4$ . This time the y– value is a constant, so in this equation, $y$ does not depend on $x$ . Fill in 4 for all the $y$ ’s in the table below and then choose any values for $x$ . We’ll use 0, 2, and 4 for the x-coordinates.

**$y=4$**
$x$	$y$	$\left(x,y\right)$
0	4	$\left(0,4\right)$
2	4	$\left(2,4\right)$
4	4	$\left(4,4\right)$

The graph is a horizontal line passing through the y-axis at 4. See (Figure 10).

The points listed in the previous table are plotted. The equation y = 4 is graphed. The resulting line is horizontal. — Figure .10

Horizontal line

A horizontal line is the graph of an equation of the form $y=b$ .

The line passes through the y-axis at $\left(0,b\right)$ .

EXAMPLE 8

Graph the equation $y=-1.$

Solution

The equation $y=-1$ has only one variable, $y$ . The value of $y$ is constant. All the ordered pairs in the table below have the same y-coordinate. The graph is a horizontal line passing through the y-axis at $-1$ , as shown in (Figure 11).

**$y=-1$**
$x$	$y$	$\left(x,y\right)$
0	$-1$	$\left(0,-1\right)$
3	$-1$	$\left(3,-1\right)$
$-3$	$-1$	$\left(-3,-1\right)$

The points listed in the previous table are plotted. The equation y = −1 is graphed. The resulting line is horizontal. — Figure .11

TRY IT 8

Graph the equation $y=-4$ .

Show answer

Graph of the equation y = −4. The resulting line is horizontal.

The equations for vertical and horizontal lines look very similar to equations like $y=4x.$ What is the difference between the equations $y=4x$ and $y=4$ ?

The equation $y=4x$ has both $x$ and $y$ . The value of $y$ depends on the value of $x$ . The y-coordinate changes according to the value of $x$ . The equation $y=4$ has only one variable. The value of $y$ is constant. The y-coordinate is always 4. It does not depend on the value of $x$ . See the tables below.

**$y=4x$**
$x$	$y$	$\left(x,y\right)$
0	4	$\left(0,4\right)$
2	4	$\left(2,4\right)$
4	4	$\left(4,4\right)$

**$y=4$**
$x$	$y$	$\left(x,y\right)$
0	0	$\left(0,0\right)$
1	4	$\left(1,4\right)$
2	8	$\left(2,8\right)$

The equations y = 4 and y = 4x are graphed and labelled. — Figure .12

Notice, in (Figure 12), the equation $y=4x$ gives a slanted line, while $y=4$ gives a horizontal line.

EXAMPLE 9

Graph $y=-3x$ and $y=-3$ in the same rectangular coordinate system.

Solution

Notice that the first equation has the variable $x$ , while the second does not. See the tables below. The two graphs are shown in (Figure 13).

**$y=-3x$**
$x$	$y$	$\left(x,y\right)$
0	0	$\left(0,0\right)$
1	$-3$	$\left(1,-3\right)$
2	$-6$	$\left(2,-6\right)$

**$y=-3$**
$x$	$y$	$\left(x,y\right)$
0	$-3$	$\left(0,-3\right)$
1	$-3$	$\left(1,-3\right)$
2	$-3$	$\left(2,-3\right)$

The equations y = −3 and y = −3x are graphed and labelled. The equation y = −3x is a slanted line while y = −3 is horizontal. — Figure .13

TRY IT 9

Graph $y=-4x$ and $y=-4$ in the same rectangular coordinate system.

Show answer

The equations y = −4 and y = −4x are graphed and labelled. The equation y = −4x is a slanted line while y = −4 is horizontal.

Key Concepts

Graph a Linear Equation by Plotting Points
1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work!
3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

Glossary

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.

horizontal line: A horizontal line is the graph of an equation of the form $y=b$ . The line passes through the y-axis at $\left(0,b\right)$ .

vertical line: A vertical line is the graph of an equation of the form $x=a$ . The line passes through the x-axis at $\left(a,0\right)$ .

3.2 Exercise Set

In the following exercises, for each ordered pair, decide:

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

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

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

In the following exercises, graph by plotting points.

$y=3x-1$
$y=-3x+3$
$y=x+2$
$y=-x-3$
$y=2x$
$y=3x$
$y=\dfrac{1}{2}x+2$
$y=\dfrac{4}{3}x-5$
$y=-\dfrac{2}{5}x+1$
$y=-\dfrac{3}{2}x+2$
$x+y=6$
$x+y=-3$
$x-y=2$
$x-y=-1$
$3x+y=7$
$2x+y=-3$
$\dfrac{1}{3}x+y=2$
$-\dfrac{1}{2}x-y=-3$
$2x+3y=12$
$3x-4y=12$
$x-6y=3$
$3x+y=2$

In the following exercises, graph each equation.

$x=4$
$x=-2$
$y=3$
$y=-5$
$x=\dfrac{7}{3}$
$y=-\dfrac{15}{4}$

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

$y=2x$ and $y=2$
$y=-\dfrac{1}{2}x$ and $y=-\dfrac{1}{2}$

The Stonechilds rented a motor home for one week to go on vacation. It cost them $594 plus $0.32 per mile to rent the motor home, so the linear equation $y=594+0.32x$ gives the cost, $y$ , for driving $x$ miles. Calculate the rental cost for driving 400, 800, and 1200 miles, and then graph the line.