6.3.1: Graphing Linear Equations in Two Variables

Learning Outcomes

Plot linear equations in two variables on the coordinate plane.
Use intercepts to plot lines.
Use a graphing utility to graph a linear equation on a coordinate plane.

Key words

Graph in two variables: a graph on a 2-dimensional plane
$x$ -intercept: the point where the graph crosses the $x$ -axis
$y$ -intercept: the point where the graph crosses the $y$ -axis

Graphing Linear Equations

Using Points to Plot Linear Equations

To graph a linear equation in two variables, we can plot a set of ordered pair solutions as points on a rectangular coordinate system. Its graph is called a graph in two variables. Any graph on a two-dimensional plane is a graph in two variables.

Suppose we want to graph the equation $y=2x - 1$ . We can begin by finding solutions for the equation by substituting values for $x$ into the equation and determining the resulting value of $y$ . Each pair of $x$ and $y$ -values is an ordered pair that can be plotted. The table below lists values of $x$ from –3 to 3 and the resulting values for $y$ .

$x$	$y=2x - 1$	$\left(x,y\right)$
$-3$	$y=2\left(-3\right)-1=-7$	$\left(-3,-7\right)$
$-2$	$y=2\left(-2\right)-1=-5$	$\left(-2,-5\right)$
$-1$	$y=2\left(-1\right)-1=-3$	$\left(-1,-3\right)$
$0$	$y=2\left(0\right)-1=-1$	$\left(0,-1\right)$
$1$	$y=2\left(1\right)-1=1$	$\left(1,1\right)$
$2$	$y=2\left(2\right)-1=3$	$\left(2,3\right)$
$3$	$y=2\left(3\right)-1=5$	$\left(3,5\right)$

When we plot the points in the table, they form a line, so we can connect them.

This is not true for all equations, but the graph of a linear equation is always a line.

This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).

Note that the $x$ –values chosen are arbitrary regardless of the type of equation we are graphing. Of course, some situations may require particular values of $x$ to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least 2 to graph a line and at least 3 to guarantee the line is correct. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.

How To graph a linear equation

Make a solutions table.
Plot the ordered pairs on a rectangular coordinate system.
Connect the points if they form a line.

Example

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

Solution

First, we construct a table by choosing $x$ -values and calculating the corresponding $y$ -values.

$x$	$y=-x+2$	$\left(x,y\right)$
$-5$	$y=-\left(-5\right)+2=7$	$\left(-5,7\right)$
$-3$	$y=-\left(-3\right)+2=5$	$\left(-3,5\right)$
$-1$	$y=-\left(-1\right)+2=3$	$\left(-1,3\right)$
$0$	$y=-\left(0\right)+2=2$	$\left(0,2\right)$
$1$	$y=-\left(1\right)+2=1$	$\left(1,1\right)$
$3$	$y=-\left(3\right)+2=-1$	$\left(3,-1\right)$
$5$	$y=-\left(5\right)+2=-3$	$\left(5,-3\right)$

Now, plot the points. Connect them since they form a line.

This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).

Try It

Construct a table and graph the equation by plotting points: $y=\frac{1}{2}x+2$ .

Show Answer

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

This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3/2); (0, 2); (1, 5/2); and (2, 3).

TRY IT

Using Intercepts to Plot Linear Equations

The intercepts of a graph are points where the graph crosses the axes. The ${x}$ -intercept is the point where the graph crosses the $x$ –axis. At this point, the $y$ –coordinate is zero. The The ${y}$ –intercept is the point where the graph crosses the $y$ –axis. At this point, the $x$ –coordinate is zero.

INTERCEPTS

The $x$ -intercept is the point where the graph crosses the $x$ -axis. At this point $y=0$ .

The $y$ -intercept is the point where the graph crosses the $y$ -axis. At this point $x=0$ .

To determine the $x$ –intercept, we set $y$ equal to zero and solve for $x$ . Similarly, to determine the $y$ –intercept, we set $x$ equal to zero and solve for $y$ . For example, let’s find the intercepts of the equation $y=3x - 1$ .

To find the $x$ –intercept, set $y=0$ .

\begin{array}{llllll} y = 3 x - 1 \\ 0 = 3 x - 1 \\ 1 = 3 x \\ \frac{1}{3} = x \\ (\frac{1}{3}, 0) & x -intercept \end{array}

$\begin{array}{llllll}y=3x - 1\hfill & \hfill \\ 0=3x - 1\hfill & \hfill \\ 1=3x\hfill & \hfill \\ \frac{1}{3}=x\hfill & \hfill \\ \left(\frac{1}{3},0\right)\hfill & x\text{-intercept}\hfill \end{array}$

To find the $y$ –intercept, set $x=0$ .

\begin{array}{lllll} y = 3 x - 1 \\ y = 3 (0) - 1 \\ y = - 1 \\ (0, - 1) & y -intercept \end{array}

$\begin{array}{lllll}y=3x - 1\hfill & \hfill \\ y=3\left(0\right)-1\hfill & \hfill \\ y=-1\hfill & \hfill \\ \left(0,-1\right)\hfill & y\text{-intercept}\hfill \end{array}$

We can confirm that our results make sense by looking at a graph of the equation. Notice that the graph crosses the axes where we predicted it would.

This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x – 1 is plotted on the coordinate plane