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Study Guides > College Algebra

Graph Linear Equations

Learning Objectives

  • 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−1y=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.
xx y=2x−1y=2x - 1 (x,y)\left(x,y\right)
−3-3 y=2(−3)−1=−7y=2\left(-3\right)-1=-7 (−3,−7)\left(-3,-7\right)
−2-2 y=2(−2)−1=−5y=2\left(-2\right)-1=-5 (−2,−5)\left(-2,-5\right)
−1-1 y=2(−1)−1=−3y=2\left(-1\right)-1=-3 (−1,−3)\left(-1,-3\right)
00 y=2(0)−1=−1y=2\left(0\right)-1=-1 (0,−1)\left(0,-1\right)
11 y=2(1)−1=1y=2\left(1\right)-1=1 (1,1)\left(1,1\right)
22 y=2(2)−1=3y=2\left(2\right)-1=3 (2,3)\left(2,3\right)
33 y=2(3)−1=5y=2\left(3\right)-1=5 (3,5)\left(3,5\right)
We can plot the points in 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 the graphing simpler.
  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 y=−x+2y=-x+2 by plotting points.

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

xx y=−x+2y=-x+2 (x,y)\left(x,y\right)
−5-5 y=−(−5)+2=7y=-\left(-5\right)+2=7 (−5,7)\left(-5,7\right)
−3-3 y=−(−3)+2=5y=-\left(-3\right)+2=5 (−3,5)\left(-3,5\right)
−1-1 y=−(−1)+2=3y=-\left(-1\right)+2=3 (−1,3)\left(-1,3\right)
00 y=−(0)+2=2y=-\left(0\right)+2=2 (0,2)\left(0,2\right)
11 y=−(1)+2=1y=-\left(1\right)+2=1 (1,1)\left(1,1\right)
33 y=−(3)+2=−1y=-\left(3\right)+2=-1 (3,−1)\left(3,-1\right)
55 y=−(5)+2=−3y=-\left(5\right)+2=-3 (5,−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=12x+2y=\frac{1}{2}x+2.

Answer:

xx y=12x+2y=\frac{1}{2}x+2 (x,y)\left(x,y\right)
−2-2 y=12(−2)+2=1y=\frac{1}{2}\left(-2\right)+2=1 (−2,1)\left(-2,1\right)
−1-1 y=12(−1)+2=32y=\frac{1}{2}\left(-1\right)+2=\frac{3}{2} (−1,32)\left(-1,\frac{3}{2}\right)
00 y=12(0)+2=2y=\frac{1}{2}\left(0\right)+2=2 (0,2)\left(0,2\right)
11 y=12(1)+2=52y=\frac{1}{2}\left(1\right)+2=\frac{5}{2} (1,52)\left(1,\frac{5}{2}\right)
22 y=12(2)+2=3y=\frac{1}{2}\left(2\right)+2=3 (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).

Use a Graphing Utility to Plot Lines

You can use Desmos to quickly plot lines. Watch this short video Tutorial to learn how. https://youtu.be/cEIOdi2R4fE

Try it now

Desmos has a helpful feature that allows you to turn a constant (number) into a variable. Follow these steps to learn how:
  1. Graph the line y=−23x−43y=-\frac{2}{3}x-\frac{4}{3}
  2. On the next line enter y=−ax−43y=-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.
  3. 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. https://youtu.be/9MChp2P0vMA  

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