Graphing Linear Equations

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.

We can plot a set of points to represent an equation. When such an equation contains both an x variable and a y variable, it is called an equation in two variables. 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 substituting a value 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.

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

We can plot these points from the table. The points for this particular equation form a line, so we can connect them. This is not true for all equations.

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 two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.

How To: Given an equation, graph by plotting points

Make a table with one column labeled x, a second column labeled with the equation, and a third column listing the resulting ordered pairs.
Enter x-values down the first column using positive and negative values. Selecting the x-values in numerical order will make graphing easier.
Select x-values that will yield y-values with little effort, preferably ones that can be calculated mentally.
Plot the ordered pairs.
Connect the points if they form a line.

Example: Graphing an Equation in Two Variables by Plotting Points

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

Show Solution

First, we construct a table similar to the one below. Choose x values and calculate y.

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

Now, plot the points. Connect them if 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 Show Solution

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

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).

Using Intercepts to Plot Lines in the Coordinate Plane

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 y-intercept is the point where the graph crosses the y-axis. At this point, the x-coordinate is zero.

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, lets 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 observing 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

How To: Given an equation, find the intercepts

Find the x-intercept by setting $y=0$ and solving for $x$ .
Find the y-intercept by setting $x=0$ and solving for $y$ .

Example: Finding the Intercepts of the Given Equation

Find the intercepts of the equation $y=-3x - 4$ . Then sketch the graph using only the intercepts.

Show Solution

Set

y = 0

$y=0$ to find the x-intercept.

\begin{array}{l} y = - 3 x - 4 \\ 0 = - 3 x - 4 \\ 4 = - 3 x \\ - \frac{4}{3} = x \\ (- \frac{4}{3}, 0) x -intercept \end{array}

$\begin{array}{l}y=-3x - 4\hfill \\ 0=-3x - 4\hfill \\ 4=-3x\hfill \\ -\frac{4}{3}=x\hfill \\ \left(-\frac{4}{3},0\right)x\text{-intercept}\hfill \end{array}$

Set $x=0$ to find the y-intercept.

\begin{array}{l} y = - 3 x - 4 \\ y = - 3 (0) - 4 \\ y = - 4 \\ (0, - 4) y -intercept \end{array}

$\begin{array}{l}y=-3x - 4\hfill \\ y=-3\left(0\right)-4\hfill \\ y=-4\hfill \\ \left(0,-4\right)y\text{-intercept}\hfill \end{array}$

Plot both points and draw a line passing through them.

This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4/3, 0) and (0, -4).

Try It

Find the intercepts of the equation and sketch the graph: $y=-\frac{3}{4}x+3$ .

Show Solution

x-intercept is $\left(4,0\right)$ ; y-intercept is $\left(0,3\right)$ .

This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x/4 + 3 is plotted.

Using a Graphing Utility to Plot Lines

You can use an online graphing tool to quickly plot lines. Watch this short video Tutorial to learn how.

Try It

Desmos has a helpful feature that allows you to turn a constant (number) into a variable. Follow these steps to learn how:

Graph the line $y=-\frac{2}{3}x-\frac{4}{3}$ .
On the next line enter $y=-a x-\frac{4}{3}$ . You will see a button pop up that says “add slider: a”, click on the button. You will see the next line populated with the variable a and the interval on which a can take values.
What part of a line does the variable a represent? The slope or the y-intercept?

Here is a short tutorial with more information about sliders.

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Module 3: The Rectangular Coordinate System and Equations of Lines

Learning Outcomes

How To: Given an equation, graph by plotting points

Example: Graphing an Equation in Two Variables by Plotting Points

Try It

Using Intercepts to Plot Lines in the Coordinate Plane

How To: Given an equation, find the intercepts

Example: Finding the Intercepts of the Given Equation

Try It

Using a Graphing Utility to Plot Lines

Try It

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