Find the slope from two points
key notes:
What is Slope?
- Slope tells us how steep a line is.
- It shows how much the y-value changes for every change in the x-value.
- Slope is also called the rate of change or gradient.
Formula for Slope
If two points on a line are (x₁, y₁) and (x₂, y₂), the slope (m) is:
m = y2 − y1 / x2 − x1
Read as:
Slope = Change in y ÷ Change in x
or
Slope = rise ÷ run
Steps to Find the Slope
- Label the points: (x₁, y₁) and (x₂, y₂)
- Subtract y-values: y2−y1 → this is the rise.
- Subtract x-values: x2−x1→ this is the run.
- Divide: rise / run
- Simplify the fraction if possible.
Example
Find the slope of the line passing through (2, 5) and (6, 13):
m = 13−5 / 6−2 = 8/4 = 2
Slope = 2 → For every 1 unit you move right, the line goes up by 2 units.
Types of Slope
- Positive slope: Line goes up from left to right.
- Negative slope: Line goes down from left to right.
- Zero slope: Horizontal line (no rise).
- Undefined slope: Vertical line (no run).
Common Mistakes
- Mixing up x and y values.
- Forgetting to subtract in the same order for both numerator and denominator.
- Dividing by zero (happens in vertical lines → slope is undefined).
Real-life Connection
- Slope is used in roads, ramps, roofs, and graphs to show steepness or rate of change.
Learn with an example
Find the slope of the line that passes through (9, 4) and (2, 2).
Plug (9, 4) and (2, 2) into the slope formula.
Slope = change in y / change in x
= 2-4 / 2-9
= -2/-7
= 2/7
The slope is 2/7 .
Find the slope of the line that passes through (4, 3) and (6, 4).
Plug (4, 3) and (6, 4) into the slope formula.
Slope =change in y / change in x
= 4-3 / 6-4
= 1/2
The slope is 1/2
Find the slope of the line that passes through (2, 5) and (8, 4).
Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Plug (2, 5) and (8, 4) into the slope formula.
slope = change in y / change in x
= 4 – 5 / 8 – 2 Plug in (2, 5) and (8, 4)
= -1 / 6 Subtract
The slope is -1 / 6 .
Let’s Practice!