How to Find the Slope of a Line — Formula and Examples

Grade: 8–9 | Topic: Algebra

What You Will Learn

By the end of this page, you will be able to calculate the slope of a line from two points using the slope formula, interpret what positive, negative, zero, and undefined slopes mean visually, and use slope in the slope-intercept equation $y = mx + b$.

Theory

What Is Slope?

Slope measures how steep a line is and which direction it goes. It tells you: for every 1 unit you move to the right along the x-axis, how many units does the line go up or down?

Slope is often described as rise over run:

$m = \frac{\text{rise}}{\text{run}} = \frac{\text{vertical change}}{\text{horizontal change}}$

The Slope Formula

Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

The order matters — use the same point as "point 1" for both the numerator and denominator. Swapping both is fine; swapping only one gives the wrong sign.

Types of Slope

Slope	Line goes...	Example
Positive ($m > 0$)	Uphill left to right	$m = 2$
Negative ($m < 0$)	Downhill left to right	$m = -3$
Zero ($m = 0$)	Perfectly horizontal	$y = 4$
Undefined	Perfectly vertical	$x = -1$

A vertical line has undefined slope because the denominator $x_2 - x_1 = 0$, and division by zero is undefined.

Slope in $y = mx + b$

In the slope-intercept form $y = mx + b$:

• $m$ is the slope
• $b$ is the y-intercept (where the line crosses the y-axis)

You can read the slope directly from the equation without calculating.

Worked Examples

Example 1: Slope from Two Points

Problem: Find the slope of the line passing through $(2, 3)$ and $(6, 11)$.

Step 1: Label the points: $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (6, 11)$.

Step 2: Apply the slope formula. $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$

Answer: The slope is $\mathbf{m = 2}$ (for every 1 unit right, the line goes 2 units up).

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Example 2: Negative Slope

Problem: Find the slope of the line through $(-1, 5)$ and $(3, -3)$.

Step 1: Apply the formula. $m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2$

Answer: The slope is $\mathbf{m = -2}$ (the line goes downhill).

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Example 3: Reading Slope from an Equation

Problem: What is the slope of $y = -\dfrac{3}{4}x + 7$?

Step 1: Identify $m$ in $y = mx + b$.

The coefficient of $x$ is $-\dfrac{3}{4}$.

Answer: Slope $= \mathbf{-\dfrac{3}{4}}$ (the line falls 3 units for every 4 units to the right).

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Example 4: Slope from a Table of Values

Problem: The table shows values of $x$ and $y$ for a linear relationship.

$x$	0	2	4	6
$y$	1	5	9	13

Find the slope.

Step 1: Pick any two points, e.g. $(0, 1)$ and $(2, 5)$.

$m = \frac{5 - 1}{2 - 0} = \frac{4}{2} = 2$

Step 2: Verify with another pair: $(4, 9)$ and $(6, 13)$. $m = \frac{13 - 9}{6 - 4} = \frac{4}{2} = 2 \checkmark$

Answer: Slope $= \mathbf{2}$

Common Mistakes

Mistake 1: Subtracting x-values and y-values in the Wrong Order

❌ $m = \frac{x_2 - x_1}{y_2 - y_1}$ (run over rise)

✅ $m = \frac{y_2 - y_1}{x_2 - x_1}$ (rise over run). It is always the $y$ difference on top.

Mistake 2: Mixing Up Which Point is "Point 1"

❌ $m = \frac{y_2 - y_1}{x_1 - x_2}$ (used $x_1 - x_2$ instead of $x_2 - x_1$).

✅ Be consistent — subtract in the same order top and bottom. You can use either point as point 1, but you must use the same order for both the numerator and denominator.

Mistake 3: Calling a Horizontal Line "Undefined"

❌ The line $y = 5$ has undefined slope.

✅ $y = 5$ is horizontal — slope = 0 (not undefined). A vertical line like $x = 3$ has undefined slope.

Practice Problems

Try these on your own before checking the answers:

1. Find the slope through $(1, 4)$ and $(5, 12)$.
2. Find the slope through $(3, 7)$ and $(3, -2)$.
3. Find the slope through $(-2, 6)$ and $(4, 6)$.
4. What is the slope of $y = 5x - 3$?
5. A ramp rises 1.2 metres over a horizontal distance of 4 metres. What is its slope?

Click to see answers

1. $m = \frac{12 - 4}{5 - 1} = \frac{8}{4} = \mathbf{2}$
2. $m = \frac{-2 - 7}{3 - 3} = \frac{-9}{0}$ = undefined (vertical line)
3. $m = \frac{6 - 6}{4 - (-2)} = \frac{0}{6} = \mathbf{0}$ (horizontal line)
4. Slope $= \mathbf{5}$ (coefficient of $x$)
5. $m = \frac{1.2}{4} = \mathbf{0.3}$

Summary

• Slope formula: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ — always $y$ on top (rise), $x$ on bottom (run).
• Positive slope = uphill; negative slope = downhill; zero = horizontal; undefined = vertical.
• In $y = mx + b$, the slope $m$ can be read directly from the equation.
• Always subtract in a consistent order — same point as "point 1" for both numerator and denominator.

Related Topics

• How to Graph Linear Equations
• Linear Equations — How to Solve Step by Step
• The Coordinate Plane — Plotting Points

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