Slope
The slope tells you the rate of change and whether the line rises, falls, or remains horizontal as x increases.
Free Algebra Guide
Slope-Intercept Form: y = mx + b
Learn what m and b mean, how to identify slope and y-intercept, how to graph y = mx + b, and how to write linear equations in slope-intercept form.
Slope-intercept form is one of the most useful ways to write a linear equation. When an equation is written as y = mx + b, you can quickly identify the line's slope and y-intercept. Those two pieces of information tell you how the line changes and where it crosses the y-axis.
More importantly, slope-intercept form connects three different ways of thinking about a linear relationship: its equation, its graph, and its rate of change. Learning to move between those representations is a major Algebra 1 skill.
What Is Slope-Intercept Form?
Slope-intercept form is a way of writing a linear equation so that the slope and y-intercept are immediately visible.
Slope-Intercept Form
y = mx + b
Each part of the equation has a specific meaning. The variable x is the input, y is the output, m is the slope, and b is the y-intercept.
Y-Intercept
The y-intercept tells you where the line crosses the y-axis. Its coordinate is always (0, b).
Example: y = 3x − 4
Compare the equation y = 3x − 4 with y = mx + b.
The coefficient of x is 3, so m = 3. The constant is −4, so b = −4. Therefore the line has slope 3 and crosses the y-axis at (0, −4).
| Part | Meaning | In y = 3x − 4 | What It Tells You |
|---|---|---|---|
| m | Slope | 3 | Rise 3 for every run of 1 |
| b | Y-intercept | −4 | Cross the y-axis at (0, −4) |
The key idea: in y = mx + b, m tells you how the
line moves and b tells you where the line starts
on the y-axis.
What Does m Mean in y = mx + b?
The value of m is the slope of the line. Slope tells you how much y changes when x changes.
If the slope is positive, the line rises from left to right. If the slope is negative, the line falls. If the slope is 0, the line is horizontal.
m = 2
Rise 2 units for every run of 1 unit.
m = −3
Fall 3 units for every run of 1 unit.
m =
2
5
Rise 2 units for every run of 5 units.
m = 0
The y-value does not change as x changes.
Think of m as a Rate
If m = 4, then each increase of 1 in x produces an increase of 4 in y. If m = −2, each increase of 1 in x produces a decrease of 2 in y.
What Does b Mean in y = mx + b?
The value of b is the y-intercept. It tells you where the line crosses the y-axis.
Every point on the y-axis has an x-coordinate of 0. That is why the y-intercept is always written as the ordered pair (0, b).
Example
y = 2x + 5
b = 5 → y-intercept = (0, 5)
b = 5 → y-intercept = (0, 5)
1. Find the constant
Look for the number that is not multiplied
by x.
2. That number is b
In y = 2x + 5,
b = 5.
3. Turn b into a point
Since the y-axis has x = 0, the point is
(0, 5).
Why Does b Equal the Y-Intercept?
Substitute x = 0 into y = mx + b:
y = m(0) + b = b
So whenever x = 0, y = b. That gives the point (0, b).
How to Identify m and b
When a linear equation is already written in slope-intercept form, identifying m and b is usually very quick. Compare the equation directly with y = mx + b.
Example 1
y = 4x − 7
m = 4 b = −7
m = 4 b = −7
Example 2
y = −x + 3
m = −1 b = 3
m = −1 b = 3
Example 3
y =
2
3
x − 5
m = 2/3 b = −5
m = 2/3 b = −5
Watch for Invisible Coefficients
In y = x + 6, the coefficient of x is 1, so m = 1.
In y = −x + 6, the coefficient is −1, so m = −1.
| Equation | m | b | Y-Intercept |
|---|---|---|---|
| y = 5x + 2 | 5 | 2 | (0, 2) |
| y = −3x + 4 | −3 | 4 | (0, 4) |
| y = x − 8 | 1 | −8 | (0, −8) |
| y = −x | −1 | 0 | (0, 0) |
How to Graph Slope-Intercept Form
Slope-intercept form makes graphing especially convenient because the equation tells you exactly where to begin and how to move.
Consider the equation y = 2x − 3.
Equation
y = 2x − 3
Y-Intercept
(0, −3)
Slope
2 = 2/1
1. Plot the y-intercept
Since b = −3, begin at
(0, −3).
2. Rewrite the slope as rise/run
m = 2 can be written as
2/1.
3. Use the rise
Move up 2 units.
4. Use the run
Move right 1 unit. You reach
(1, −1).
5. Draw the line
Draw a straight line through the two
points and extend it in both directions.
Graphing y = mx + b is easiest when you remember:
start at b, then move using m.
How to Write an Equation From m and b
If you know the slope and y-intercept, you can write the equation by substituting those values directly into y = mx + b.
Given
m = 3
b = −2
1. Start with the formula
Write
y = mx + b.
2. Replace m
Substitute 3 for m:
y = 3x + b.
3. Replace b
Substitute −2 for b:
y = 3x + (−2).
4. Simplify the signs
The final equation is
y = 3x − 2.
Be Careful With Negative b
If b is negative, do not write + −2 in the final equation. Simplify the signs so y = 3x + (−2) becomes y = 3x − 2.
How to Write a Slope-Intercept Equation From a Graph
When you are given a graph instead of an equation, you can work backward. Find the y-intercept and the slope, then place those values into y = mx + b.
1. Find the y-intercept
Look for the point where the line crosses
the y-axis.
2. Record b
If the line crosses at
(0, 4), then
b = 4.
3. Find another point
Choose a second clear grid point on the
line.
4. Calculate rise over run
Suppose you move down 2 and right 1.
Then m = −2.
5. Write the equation
Substitute m = −2 and b = 4:
y = −2x + 4.
Why Find b First?
The y-intercept is often the easiest exact point to read from a graph. Once you know b, you only need the slope to finish the equation.
Fractional and Negative Slopes in y = mx + b
The slope in slope-intercept form does not have to be a whole number. Fractions are often especially useful because they show the rise and run directly.
y =
2
3
x + 1
Start at (0, 1), then rise 2 and run 3.
y = −
3
4
x + 5
Start at (0, 5), then fall 3 and run 4.
A Negative Sign Can Be Placed in Different Parts
The slope −3/4 can be interpreted as:
- down 3 and right 4, or
- up 3 and left 4.
Special Cases in Slope-Intercept Form
Some slope-intercept equations look slightly different because a coefficient or constant is not written explicitly.
| Equation | m | b | Meaning |
|---|---|---|---|
| y = x + 3 | 1 | 3 | The invisible coefficient of x is 1 |
| y = −x + 3 | −1 | 3 | The invisible coefficient is −1 |
| y = 4x | 4 | 0 | The line crosses at the origin |
| y = 6 | 0 | 6 | Horizontal line through (0, 6) |
Vertical Lines Are Not in Slope-Intercept Form
A vertical line such as x = 4 has undefined slope. Because slope-intercept form uses a finite slope m, vertical lines cannot be written as y = mx + b.
Worked Slope-Intercept Form Examples
y = 5x − 2
- Compare with y = mx + b.
- The coefficient of x is 5.
- Therefore m = 5.
- The constant is −2, so b = −2.
- The y-intercept is (0, −2).
m = −3, b = 7
- Start with y = mx + b.
- Substitute m = −3.
- Substitute b = 7.
- y = −3x + 7.
m = 2/5, b = −1
- Start at the y-intercept (0, −1).
- Use slope 2/5.
- Rise 2 and run 5.
- The equation is y = (2/5)x − 1.
Crosses at (0, 4), slope = −2
- The y-intercept gives b = 4.
- The slope gives m = −2.
- Substitute into y = mx + b.
- y = −2x + 4.
y = 3x
- The coefficient gives m = 3.
- No constant is written.
- Therefore b = 0.
- The line passes through the origin.
y = −4
- There is no x-term.
- Therefore m = 0.
- The constant gives b = −4.
- The graph is a horizontal line through (0, −4).
Common Slope-Intercept Form Mistakes
Mixing Up m and b
In y = mx + b, m is the slope and b is the y-intercept. The coefficient of x is not the y-intercept.
Forgetting the Sign of b
In y = 2x − 5, b = −5, not 5. The line crosses the y-axis at (0, −5).
Missing an Invisible 1
In y = x + 4, m = 1. In y = −x + 4, m = −1.
Treating b as an X-Intercept
The value b gives the y-intercept, not the x-intercept. The corresponding point is always (0, b).
Using Run Over Rise
The slope is rise over run. For a slope of 2/3, move vertically 2 units and horizontally 3 units.
Starting With the Slope Before Plotting b
When graphing y = mx + b, plot the y-intercept first. Then use the slope from that point.
Thinking Every Line Has Slope-Intercept Form
Vertical lines such as x = 3 have undefined slope and cannot be written in the form y = mx + b.
A Reliable Strategy for Slope-Intercept Form
1. Look for y = mx + b
Determine whether the equation is already
written in slope-intercept form.
2. Identify m
Find the coefficient multiplying x.
That value is the slope.
3. Identify b
Find the constant term. That value gives
the y-intercept (0, b).
4. Interpret the slope
Read m as rise over run and check whether
the line should rise, fall, or remain
horizontal.
5. Graph or write the equation
If graphing, start at b and move using m.
If writing an equation, substitute both
values into y = mx + b.
6. Check the result
Verify that the graph crosses the y-axis
at the correct point and that its direction
agrees with the sign of the slope.
A quick check catches many mistakes:
does the line cross at (0, b), and does its
direction match the sign of m?
Practice Slope-Intercept Form
Practice identifying m and b, interpreting slope and y-intercept, building equations, and matching equations to graphs with step-by-step feedback.
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