Slope-Intercept Form
Slope-intercept form is a way to write the equation of a line: y = mx + b. The m is the slope, which tells how steep the line is. The b is the y-intercept, where the line crosses the y-axis. With this form, you can graph a line in two quick steps.
Before this lesson: You should know how to find slope and what the y-intercept of a line is.
What Is Slope-Intercept Form?
Slope-intercept form names the two things you need to draw a line: where it starts and how it moves.
y = mx + b \qquad m = \text{slope}, \quad b = \text{y-intercept}- The slope m tells you the rise over the run.
- The y-intercept b is the y-value where the line crosses the y-axis. That point is (0, b).
Here are some equations in slope-intercept form.
| Equation | Slope m | y-intercept b |
|---|---|---|
| y = 2x + 3 | 2 | 3 |
| y = -x + 4 | −1 | 4 |
| y = \frac{1}{2}x - 5 | \frac{1}{2} | −5 |
| y = 3x | 3 | 0 |
| y = -6 | 0 | −6 |
Tip: b keeps its sign. In y = 4x - 7, the y-intercept is −7, not 7.
How to Graph a Line in Slope-Intercept Form
- Plot the y-intercept (0, b) on the y-axis.
- Write the slope as a fraction, rise over run.
- From the y-intercept, move up or down by the rise, then right by the run. Plot a second point.
- Draw a straight line through the two points.
If the slope is a whole number, write it over 1. A slope of 3 is \frac{3}{1}: up 3, right 1. A slope of −2 is \frac{-2}{1}: down 2, right 1.
Worked Examples
Example 1: Find the slope and y-intercept
What are the slope and y-intercept of y = -3x + 5?
Step 1. The number in front of x is −3, so m = -3.
Step 2. The number added at the end is 5, so b = 5.
Answer: The slope is −3, and the line crosses the y-axis at (0, 5).
Example 2: Graph a line with a fraction slope
Graph y = \frac{2}{3}x - 1.
Step 1. The y-intercept is −1. Plot (0, −1).
Step 2. The slope is \frac{2}{3}, so the rise is 2 and the run is 3.
Step 3. From (0, −1), go up 2 and right 3. Plot (3, 1).
Step 4. Draw the line through (0, −1) and (3, 1). The graph above shows this line.
Example 3: Graph a line with a negative slope
Graph y = -2x + 4.
Step 1. Plot the y-intercept (0, 4).
Step 2. The slope is -2 = \frac{-2}{1}. From (0, 4), go down 2 and right 1 to reach (1, 2).
Step 3. Repeat to get (2, 0). Draw the line through the points.
Check: The line goes down from left to right, which matches a negative slope.
Example 4: Rewrite an equation in slope-intercept form
Find the slope and y-intercept of 4x - 2y = 8.
Step 1. Get y by itself. Subtract 4x from both sides, then divide every term by −2.
\begin{aligned}
4x - 2y &= 8 \\
-2y &= -4x + 8 \\
y &= 2x - 4
\end{aligned}Answer: y = 2x - 4. The slope is 2, and the y-intercept is −4.
Example 5: Word problem
A gym charges a $25 sign-up fee plus $10 per month. Write an equation for the total cost C after t months. What do the slope and y-intercept mean? How much do 6 months cost?
Step 1. The cost goes up $10 each month, so the slope is 10.
Step 2. You pay $25 before any months pass, so the y-intercept is 25.
Step 3. Write the equation: C = 10t + 25.
Step 4. For 6 months, put in t = 6: C = 10(6) + 25 = 85.
Answer: C = 10t + 25. The slope is $10 per month, and the y-intercept is the $25 sign-up fee. Six months cost $85.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Dropping the sign of b | y = 3x - 2, so b = 2 | b keeps its sign: b = -2. |
| Plotting b on the wrong axis | Starting at (−2, 0) for b = -2 | The y-intercept is on the y-axis: (0, −2). |
| Swapping rise and run | Slope \frac{2}{3} drawn as up 3, right 2 | The top number is the rise: up 2, right 3. |
| Going the wrong way | Slope −2 drawn as up 2, right 1 | A negative slope goes down: down 2, right 1. |
| Reading m before solving for y | 2x + y = 5, so m = 2 | Solve for y first: y = -2x + 5, so m = -2. |
Special Cases
- Lines through the origin: If b = 0, the equation is y = mx. The line passes through (0, 0).
- Horizontal lines: If m = 0, the equation is y = b. The line is flat.
- Vertical lines: A vertical line like x = 3 has no slope, so it can’t be written in slope-intercept form. See Horizontal and Vertical Lines.
Practice Problems
For problems 1–7, give the slope and y-intercept. Rewrite the equation in slope-intercept form first if you need to.
- y = 4x - 9
- y = -x + 6
- y = \frac{2}{5}x
- y = 8
- 3x + y = 2
- 6x - 3y = 12
- x + 2y = 10
- Graph y = 3x - 2. Name the y-intercept and two more points on the line.
- Graph y = -\frac{3}{4}x + 3. Name the y-intercept and one more point on the line.
- A candle is 12 inches tall. It burns down 2 inches every hour. Write an equation for its height h after t hours. When does it burn out?
Show answers
- m = 4, b = -9
- m = -1, b = 6
- m = \frac{2}{5}, b = 0. The line goes through the origin.
- m = 0, b = 8. This is a horizontal line.
- Subtract 3x: y = -3x + 2. So m = -3 and b = 2.
- -3y = -6x + 12, so y = 2x - 4. So m = 2 and b = -4.
- 2y = -x + 10, so y = -\frac{1}{2}x + 5. So m = -\frac{1}{2} and b = 5.
- The y-intercept is (0, −2). Go up 3 and right 1 to get (1, 1), then (2, 4).
- The y-intercept is (0, 3). Go down 3 and right 4 to get (4, 0).
- h = -2t + 12. The slope is −2 inches per hour, and the candle starts at 12 inches. It burns out when h = 0: 0 = -2t + 12, so t = 6. It burns out after 6 hours.
Slope-Intercept Form FAQ
What does b stand for in y = mx + b?
In y = mx + b, b is the y-intercept. It is the y-value where the line crosses the y-axis, at the point (0, b).
Why is it called slope-intercept form?
It is called slope-intercept form because the equation shows the slope m and the y-intercept b right away. You don’t have to do any work to read them.
Can every line be written in slope-intercept form?
No. Vertical lines like x = 3 have an undefined slope, so they can’t be written as y = mx + b. Every other line can.
How do you write slope-intercept form from two points?
Find the slope from the two points, then use one point to solve for b. The next lesson, Writing Equations in Slope-Intercept Form, shows every step.
Summary
- Slope-intercept form is y = mx + b.
- m is the slope, and b is the y-intercept at (0, b).
- To graph, plot (0, b), then use rise over run to find a second point.
- Solve for y before you read m and b.
- Vertical lines can’t be written in this form.
Previous lesson: Horizontal and Vertical Lines
Next lesson: Writing Equations in Slope-Intercept Form
Back to the Algebra 1 Index