3.5 Slope-Intercept Form of a Line

Scott McDaniel

3.5 Slope-Intercept Form of a Line

Learning Objectives

After completing this section, you should be able to:

Recognize the relationship between a line’s graph and its equation in slope-intercept form.
Identify the slope and y-intercept from an equation.
Graph a line using its slope and intercept.
Write an equation in slope-intercept form from given information.

The Slope-Intercept Form

The slope-intercept form of a line is:

$y = mx + b$

where:

m is the slope (rise/run), and
b is the y-intercept (where the line crosses the y-axis).

This form is very useful because it gives both the slope and intercept directly.

Example 3.5.1 — Identify Slope and y-Intercept

Find the slope and y-intercept of each equation.

a) $y = 2x + 3$

Solution

Slope: $m = 2$

y-intercept: $(0,3)$

b) $y = -4x + 1$

Solution

Slope: $m = -4$

y-intercept: $(0,1)$

c) $y = \frac{1}{2}x - 6$

Solution

Slope: $m = \frac{1}{2}$

y-intercept: $(0,-6)$

Try It

Finding Slope and Intercept from Standard Form

If the equation is not already in slope-intercept form, solve for y.

Example 3.5.2 — Convert to Slope-Intercept Form

Convert $2x + y = 5$ to slope-intercept form, then identify the slope and y-intercept.

Solution

Subtract $2x$ from both sides:

$y = -2x + 5$

Slope: $-2$

y-intercept: $(0,5)$

Example 3.5.3 — Convert to Slope-Intercept Form

Convert $3x - 2y = 6$ to slope-intercept form, then identify the slope and y-intercept.

Solution

Subtract $3x$ from both sides:

$-2y = -3x + 6$

Divide by $-2$ :

$y = \frac{3}{2}x - 3$

Slope: $\frac{3}{2}$

y-intercept: $(0,-3)$

Try It

Graphing a Line Using Slope and Intercept

Plot the y-intercept $(0,b)$ .
From that point, use the slope (rise/run) to find another point.
Draw the line through the two points.

Example 3.5.4 — Graph using slope and y-intercept

Graph $y=2x+1$

Step 1: Plot the y-intercept

The y-intercept is $(0,1)$ . Plot the point $(0,1)$ on the y-axis.

Step 2: Use the slope to find another point

The slope is $m = 2$ , which means rise 2 and run 1.

From $(0,1)$ , move up 2 and right 1 to reach $(1,3)$ .

Step 3: Draw the line

Draw a straight line through $(0,1)$ and $(1,3)$ .

Line with a y-int = (0,1) and slope of 2 — **Figure 3.5A**

Example 3.5.5

Graph $y = -\frac{1}{2}x + 4$

Step 1: Plot the y-intercept

The y-intercept is $(0,4)$ . Plot the point $(0,4)$ on the y-axis.

Step 2: Use the slope to find another point

The slope is $m = -\frac{1}{2}$ , which means down 1 and right 2.

From $(0,4)$ , move down 1 and right 2 to reach $(2,3)$ .

Step 3: Draw the line

Draw a straight line through $(0,4)$ and $(2,3)$ .

Line with slope of -2 through y-int (0,4) and (2,3) — **Figure 3.5B**

Try It

Writing the Equation from a Graph

You can write the equation of a line directly from a graph by reading its slope and y-intercept.

Example 3.5.6

Write the equation of the line from the graph below.

Line with y-int = (0,-2) and through another point (1,1) — **Figure 3.5C**

Solution

The line crosses the y-axis at $(0,-2)$ , so $b = -2$ .

From the graph, the line rises 3 for every 1 to the right, so $m = 3$ .

Equation: $y = 3x - 2$

Tip: Start by finding the y-intercept, then locate another clear grid point to compute slope.

Try It

Writing the Equation from Given Information

If you are given the slope and y-intercept, substitute directly into $y = mx + b$ .

Example 3.5.7

Write the equation with slope $-4$ and y-intercept $(0,-2)$ .

Solution

$y = -4x - 2$

We often are not given the y-intercept, but just another point. In this case, we substitute that point for x and y and solve for b. Note the example below.

Example 3.5.8