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Using the Slope-Intercept Form of an Equation of a Line

Choose the Most Convenient Method to Graph a Line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?

While we could plot points, use the slope–intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let's look for some patterns to help determine the most convenient method to graph a line.

Here are six equations we graphed in this chapter, and the method we used to graph each of them.

Equation Method
#1 \(x=2\) Vertical line
#2 \(y=4\) Horizontal line
#3 \(-x+2 y=6\) Intercepts
#4 \(4 x-3 y=12\) Intercepts
#5 \(y=4 x-2\) Slope-intercept
#6 \(y=-x+4\) Slope-intercept


Equations #1 and #2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #3 and #4, both x and y are on the same side of the equation. These two equations are of the form \(Ax+By=C\). We substituted \(y=0\) to find the x-intercept and \(x=0\) to find the \( y\)-intercept, and then found a third point by choosing another value for \(x\) or \(y\).

Equations #5 and #6 are written in slope–intercept form. After identifying the slope and y-intercept from the equation we used them to graph the line.

This leads to the following strategy.

Strategy for Choosing the Most Convenient Method to Graph a Line

Consider the form of the equation.

  • If it only has one variable, it is a vertical or horizontal line.
    • \(x=a\) is a vertical line passing through the \(x\)-axis at \(a\).
    • \(y=b\) is a horizontal line passing through the \(y\)-axis at \(b\).
  • If \(y\) is isolated on one side of the equation, in the form \(y=mx+b\), graph by using the slope and \(y\)-intercept.
    • Identify the slope and \(y\)-intercept and then graph.
  • If the equation is of the form \(Ax+By=C\), find the intercepts.
    • Find the \(x\)- and \(y\)-intercepts, a third point, and then graph.

Example 4.48

Determine the most convenient method to graph each line.

  1. \(y=−6\)
  2. \(5x−3y=15\)
  3. \(x=7\)
  4. \(y=\frac{2}{5} x-1\).
Solution
  1. \(y=−6\)

    This equation has only one variable,y. Its graph is a horizontal line crossing the \(y\)-axis at \9−6\).

  2. \(5x−3y=15\)

    This equation is of the form \(Ax+By=C\). The easiest way to graph it will be to find the intercepts and one more point.

  3. \(x=7\)

    There is only one variable, \(x\). The graph is a vertical line crossing the \(x\)-axis at \(7\).

  4. \(y=\frac{2}{5} x-1\)
    Since this equation is in \(y=mx+b\) form, it will be easiest to graph this line by using the slope and y-intercept.
Try It 4.95

Determine the most convenient method to graph each line: 

  1. \(3x+2y=12\)
  2. \(y=4\)
  3. \(y=\frac{1}{5} x-4\)
  4. \(x=−7\).
Try It 4.96

Determine the most convenient method to graph each line: 

  1. \(x=6\)
  2.  \(y=-\frac{3}{4} x+1\)
  3. \(y=−8\)
  4. \(4x−3y=−1\).