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

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Course: MA007: Algebra
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Introduction

Source: OpenStax, https://openstax.org/books/elementary-algebra/pages/4-5-use-the-slope-intercept-form-of-an-equation-of-a-line
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Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we'll have one more method we can use to graph lines.

In Graph Linear Equations in Two Variables, we graphed the line of the equation \(y=\frac{1}{2} x+3\) by plotting points. See Figure 4.24. Let's find the slope of this line.

This figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line is labeled with the equation y equals one half x, plus 3. The points (0, 3), (2, 4) and (4, 5) are labeled also. A red vertical line begins at the point (2, 4) and ends one unit above the point. It is labeled “Rise equals 1”. A red horizontal line begins at the end of the vertical line and ends at the point (4, 5). It is labeled “Run equals 2. The red lines create a right triangle with the line y equals one half x, plus 3 as the hypotenuse.

Figure 4.24

The red lines show us the rise is \(1\) and the run is \(2\). Substituting into the slope formula:

\( \begin{align} \begin{array}{l} m=\frac{\text { rise }}{\operatorname{run}} \\ m=\frac{1}{2} \end{array} \end{align}\)

What is the \(y\)-intercept of the line? The \(y\)-intercept is where the line crosses the \(y\)-axis, so \(y\)-intercept is \(( 0,3)\). The equation of this line is:

\(y=\frac{1}{2} x+3\)

Notice, the line has:

\( \text { slope } m=\frac{1}{2} \text { and } y \text { -intercept }(0,3)\)

When a linear equation is solved for y , the coefficient of the x term is the slope and the constant term is the y-coordinate of the y-intercept. We say that the equation y =1 2 x +3 is in slope–intercept form.

\(m=\frac{1}{2} ; y \text { -intercept is }(0,3)\)

\(\begin{align} \begin{array}{l} y=\frac{1}{2} x+3 \\ y=m x+ b \end{array} \end{align}\)

Slope-Intercept Form of an Equation of a Line

The slope–intercept form of an equation of a line with slope \(m\) and \(y\)-intercept, \(( 0,b)\) is,

\(y=m x+b\)

Sometimes the slope–intercept form is called the "y-form".

Example 4.40

Use the graph to find the slope and y-intercept of the line, \(y=2x+1\).

Compare these values to the equation \(y=mx+b\).

Solution

To find the slope of the line, we need to choose two points on the line. We’ll use the points \(( 0,1)\)  and \((1,3)\).

  to choose two points on the line. We’ll use the points (0,1) and (1,3).
Find the rise and run. \(m=\frac{\text { rise }}{\text { run }}\)
  \(m=\frac{2}{1}\)
  \(m=2\)
Find the y-intercept of the line. The \(y\)-intercept is the point \((0,1)\).
We found slope \(m=2\) and \(y\) -intercept \((0,1)\).
 \(y=2 x+1\)
\(y=m x+b\)


The slope is the same as the coefficient of x  and the y-coordinate of the y-intercept is the same as the constant term.

Try It 4.79

Use the graph to find the slope and y-intercept of the line \(y=\frac{2}{3} x-1\). Compare these values to the equation \(y=m x+b\).

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of

Try It 4.80

Use the graph to find the slope and y-intercept of the line \(y=\frac{1}{2} x+3\). Compare these values to the equation \(y =m x +b\).

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of

Identify the Slope and y-Intercept From an Equation of a Line

In Understand Slope of a Line, we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the \(y\)-intercept as the point, and then count out the slope from there. Let's practice finding the values of the slope and y-intercept from the equation of a line.

Example 4.41

Identify the slope and y-intercept of the line with equation \(y=−3x+5\).

Solution

We compare our equation to the slope–intercept form of the equation.

\(y=m x+b\)
Write the equation of the line. \(y=-3 x+5\)
Identify the slope. \(m=-3\)
Identify the y-intercept. \(\text{y-intercept is 0,5}\)
Try It 4.81

Identify the slope and y-intercept of the line \(y=\frac{2}{5} x-1\).

Try It 4.82

Identify the slope and y-intercept of the line \(y=-\frac{4}{3} x+1\).

When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for \(y\).

Example 4.42

Identify the slope and y-intercept of the line with equation \(x+2y=6\).

Solution

This equation is not in slope–intercept form. In order to compare it to the slope–intercept form we must first solve the equation for \(y\).

Solve for \(y\). \(x +2 y = 6\)
Subtract \(x\) from each side. \(2 y=-x+6\)
Divide both sides by 2. \(\frac{2 y}{2}=\frac{-x+6}{2}\)
Simplify. \(\frac{2 y}{2}=\frac{-x}{2}+\frac{6}{2}\)
\(\left(\right.\) Remember: \(\left.\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\right)\)
Simplify. \(y=-\frac{1}{2} x+3\)
Write the slope–intercept form of the equation of the line. \(y=m x+b\)
Write the equation of the line. \(y=-\frac{1}{2} x+3\)
Identify the slope. \(m=-\frac{1}{2}\)
Identify the \(y\)-intercept. \(\text{y-intercept is 0,3}\)
Try It 4.83

Identify the slope and y-intercept of the line \(x +4 y = 8 \).

Try It 4.84

Identify the slope and y-intercept of the line \(3 x +2 y = 12 \).

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

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. \(4x−3y=−1\).