Introduction

The slope of a line tells us how steep the line is. If you already know how to plot points and graph linear equations, slope is the next natural idea: it measures how much a line goes up or down as you move left or right.

In this article, we explore:

• What “rise over run” means
• How to compute slope from two points
• How to interpret positive, negative, zero, and undefined slopes
• How slope connects to linear equations

What Is Slope?

Slope measures how a line changes vertically compared to how it changes horizontally.

• Rise = vertical change
• Run = horizontal change
• Slope formula: $$m = \frac{\text{rise}}{\text{run}}$$

If you move from one point on a line to another:

• Rise = change in $y$
• Run = change in $x$

Slope Between Two Points

Suppose you have two points:

• $A(x_1, y_1)$
• $B(x_2, y_2)$

Then the slope is: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Key ideas:

• Subtract in the same order (top and bottom).
• A positive slope means the line rises as you move right.
• A negative slope means the line falls as you move right.
• If $x_2 - x_1 = 0$, the slope is undefined (vertical line).
• If $y_2 - y_1 = 0$, the slope is 0 (horizontal line).

Interpreting Slope

• Positive slope: line goes upward
• Negative slope: line goes downward
• Zero slope: flat, horizontal
• Undefined slope: vertical line

Examples:

• $m = 2$ means “up 2 for every right 1”
• $m = -\frac{1}{3}$ means “down 1 for every right 3”

Slope and Linear Equations

A line in slope-intercept form is: $$y = mx + b$$

• $m$ is the slope
• $b$ is the $y$-intercept

Knowing the slope helps you:

• Graph the line quickly
• Understand how the line behaves
• Compare different lines

Exercises

1. Compute the slope between the points $(2,5)$ and $(6,9)$.

   Solution

   Slope: $$m = \frac{9 - 5}{6 - 2} = \frac{4}{4} = 1$$
2. Find the slope of the line passing through $(-3,4)$ and $(1,4)$.

   Solution

   Slope: $$m = \frac{4 - 4}{1 - (-3)} = \frac{0}{4} = 0$$ Horizontal line.
3. Determine the slope between $(0,0)$ and $(5,-10)$.

   Solution

   Slope: $$m = \frac{-10 - 0}{5 - 0} = \frac{-10}{5} = -2$$
4. True or false: A line with slope $0$ is horizontal.

   Solution

   Answer: True.
   A slope of $0$ means no vertical change → horizontal line.
5. Compute the slope between $(4,1)$ and $(4,7)$. What type of line is this?

   Solution

   Slope: $$m = \frac{7 - 1}{4 - 4} = \frac{6}{0}$$ Undefined slope → vertical line.
6. A line rises $6$ units when it runs $3$ units to the right. What is its slope?

   Solution

   Slope: $$m = \frac{6}{3} = 2$$
7. Find the slope of the line through $(1,-2)$ and $(4,7)$.

   Solution

   Slope: $$m = \frac{7 - (-2)}{4 - 1} = \frac{9}{3} = 3$$