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 [latex]y=2x - 1[/latex]. 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.

[latex]x[/latex] [latex]y=2x - 1[/latex] [latex]\left(x,y\right)[/latex]
[latex]-3[/latex] [latex]y=2\left(-3\right)-1=-7[/latex] [latex]\left(-3,-7\right)[/latex]
[latex]-2[/latex] [latex]y=2\left(-2\right)-1=-5[/latex] [latex]\left(-2,-5\right)[/latex]
[latex]-1[/latex] [latex]y=2\left(-1\right)-1=-3[/latex] [latex]\left(-1,-3\right)[/latex]
[latex]0[/latex] [latex]y=2\left(0\right)-1=-1[/latex] [latex]\left(0,-1\right)[/latex]
[latex]1[/latex] [latex]y=2\left(1\right)-1=1[/latex] [latex]\left(1,1\right)[/latex]
[latex]2[/latex] [latex]y=2\left(2\right)-1=3[/latex] [latex]\left(2,3\right)[/latex]
[latex]3[/latex] [latex]y=2\left(3\right)-1=5[/latex] [latex]\left(3,5\right)[/latex]

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

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

Example: Graphing an Equation in Two Variables by Plotting Points

Graph the equation [latex]y=-x+2[/latex] by plotting points.

Try It

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

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 [latex]y=3x - 1[/latex].

To find the x-intercept, set [latex]y=0[/latex].

[latex]\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}[/latex]

To find the y-intercept, set [latex]x=0[/latex].

[latex]\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}[/latex]

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

  1. Find the x-intercept by setting [latex]y=0[/latex] and solving for [latex]x[/latex].
  2. Find the y-intercept by setting [latex]x=0[/latex] and solving for [latex]y[/latex].

Example: Finding the Intercepts of the Given Equation

Find the intercepts of the equation [latex]y=-3x - 4[/latex]. Then sketch the graph using only the intercepts.

Try It

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

Using a Graphing Utility to Plot Lines

https://www.desmos.com/calculator

https://www.mathway.com/Graph

https://www.symbolab.com/graphing-calculator

Try it now

These graphing utilities have features that allow you to turn a constant (number) into a variable. Follow these steps to learn how:

  1. Graph the line [latex]y=-\frac{2}{3}x-\frac{4}{3}[/latex].
  2. On the next line enter [latex]y=-a x-\frac{4}{3}[/latex]. 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.
  3. What part of a line does the variable a represent? The slope or the y-intercept?