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

Graph a Line Using its Slope and Intercept

Now that we know how to find the slope and y-intercept of a line from its equation, we can graph the line by plotting the y-intercept and then using the slope to find another point.

Example 4.43

How to Graph a Line Using its Slope and Intercept

Graph the line of the equation  \(y=4 x −2\)  using its slope and y-intercept.

Solution

Step 1. Find the slopeintercept form of the equation. This equation is in slope-intercept form. \(y=4x-2\)
Step 2. Identify the slope and \(y\) -intercept. Use \(y=m x+b\)
Find the slope.
Find the \(y\) -intercept.		 \(\begin{align} \begin{array}{l}
y=m x+b \\ y=4 x+(-2) \\
m=4 \\ b=-2,(0,-2) \end{array} \end{align}\)
Step 3. Plot the \(y\) intercept. Plot \((0.-2)\) Plot the y intercept.
Step 4. Use the slope formula \(m=\frac{\text { rise }}{\text { run }}\) to identify the rise and the run. Identify the rise and the run. \( \begin{align}\begin{array}{r}
m=4 \\ \frac{\text { rise }}{\text { run }}=\frac{4}{1} \\ \text { rise }=4 \\ \text { run }=1 \end{array} \end{align}\)
Step 5. Starting at the \(y\) -intercept, count out the rise and run to mark the second point. Start at \((0,-2)\) and count the rise and the run.
Up 4, right 1.
Step 6. Connect the points with a line. Connect the two points with a line. Step 6. Connect the points with a line.
Try It 4.85

Graph the line of the equation  \(y=4 x +1\) using its slope and y-intercept.

Try It 4.86

Graph the line of the equation  \(y=2 x −3\) using its slope and y-intercept.

HOW TO
Graph a line using its slope and y-intercept.
  1. Step 1. Find the slope-intercept form of the equation of the line.
  2. Step 2. Identify the slope and \(y\)-intercept.
  3. Step 3. Plot the \(y\)-intercept.
  4. Step 4. Use the slope formula \(m=\frac{\text { rise }}{\text { run }}\) run to identify the rise and the run.
  5. Step 5. Starting at the y-intercept, count out the rise and run to mark the second point.
  6. Step 6. Connect the points with a line.

Example 4.44

Graph the line of the equation y=−x+4 using its slope and y-intercept.

Solution
  \(y =m x +b\)
The equation is in slope–intercept form. \(y =−x +4\)
Identify the slope and \(y\)-intercept. \(m =−1\)
  y-intercept is (0, 4)
Plot the \(y\)-intercept. See graph below.
Identify the rise and the run. \(m=\frac{-1}{1}\)
Count out the rise and run to mark the second point. rise −1, run 1
Draw the line. Draw the line.
To check your work, you can find another point on the line and make sure it is a solution of the equation. In the graph we see the line goes through (4, 0).
Check.
\( \begin{align} \begin{array}{l} y=-x+4 \\ 0 \stackrel{?}{=}-4+4 \\ 0=0 \text{✓} \end{array} \end{align} \)		  
Try It 4.87

Graph the line of the equation  \(y =−x −3\) using its slope and y-intercept.

Try It 4.88

Graph the line of the equation \(y =−x −1\) using its slope and y-intercept.

Example 4.45

Graph the line of the equation y=−23x−3 using its slope and y-intercept.

Solution
  \(y =m x +b\)
 The equation is in slope-intercept form. \(y=-\frac{2}{3} x-3\)
 Identify the slope and \(y\) -intercept. \(m=-\frac{2}{3} ; y\) -intercept is \((0,-3)\)
 Plot the \(y\) -intercept.  See graph below.
 Identify the rise and the run.  
 Count out the rise and run to mark the second point.  
 Draw the line. Draw the line.
Try It 4.89

Graph the line of the equation \(y=-\frac{5}{2} x+1\) using its slope and y-intercept.

Try It 4.90

Graph the line of the equation \(y=-\frac{3}{4} x-2\) using its slope and \(y\)-intercept.

Example 4.46

Graph the line of the equation 4x−3y=12 using its slope and y-intercept.

Solution
  \(4 x-3 y=12\)
 Find the slope-intercept form of the equation. \(-3 y=-4 x+12\)
  \(-\frac{3 y}{3}=\frac{-4 x+12}{-3}\)
 The equation is now in slope-intercept form. \(y=\frac{4}{3} x-4\)
 Identify the slope and \(y\) -intercept. \(m=\frac{4}{3}\)
   \(y\) -intercept is \((0,-4)\)
 Plot the \(y\) -intercept.  See graph below.
 Identify the rise and the run; count out the rise and run to mark the second point.  
 Draw the line. Draw the line.
Try It 4.91

Graph the line of the equation  \(2 x −y =6\) using its slope and y-intercept.

Try It 4.92

Graph the line of the equation  \(3 x −2 y = 8\)  using its slope and y-intercept.

We have used a grid with x  and y  both going from about −10  to 10 for all the equations we've graphed so far. Not all linear equations can be graphed on this small grid. Often, especially in applications with real-world data, we'll need to extend the axes to bigger positive or smaller negative numbers.

Example 4.47

Graph the line of the equation  \(y=0.2 x +45\) using its slope and y-intercept.

Solution

We'll use a grid with the axes going from about \(−80\) to \(80\).

\(y =m x +b\)
The equation is in slope-intercept form.
 \(y =0.2 x +45\)
Identify the slope and \(y\) -intercept. \(m =0.2\)
   The \(y\) -intercept is \((0,45)\)
 Plot the \(y\) -intercept.  See graph below.
Count out the rise and run to mark the second point. The slope is \(m=0.2\); in fraction form this means \(m=\frac{2}{10}\). Given the scale of our graph, it would be easier to use the equivalent fraction \(m=\frac{10}{50}\).  
 Draw the line. Draw the line.
Try It 4.93

Graph the line of the equation  \(y =0.5 x +25\) using its slope and \(y\)-intercept.

Try It 4.94

Graph the line of the equation \(y =0.1 x −30\) using its slope and y-intercept.

Now that we have graphed lines by using the slope and y-intercept, let's summarize all the methods we have used to graph lines. See Figure 4.25.

The table has two rows and four columns. The first row spans all four columns and is a header row. The header is “Methods to

Figure 4.25