💡 Why Does Slope Matter?
Slope describes how steep a line is — whether it rises, falls, or stays flat. On the GED, you'll need to calculate slope from two points, identify it from a graph, and understand its sign. This lesson covers everything!
Slope measures how steep a line is. It tells us how much the line rises or falls for every one unit it moves to the right. We write slope as the letter m.
As x increases, y increases. Walking uphill takes effort — more rise for each step right.
As x increases, y stays the same. Perfectly flat road — no rise at all.
As x increases, y decreases. Walking downhill — you have to slow yourself down.
It's thought the letter m comes from the French word "monter", meaning to climb. Easy to remember!
How much the line goes up or down. Subtract the y-values: y₂ − y₁.
How far the line moves right. Subtract the x-values: x₂ − x₁.
Always use the same order for both points. If you start with point 2 for the y-values (y₂ − y₁), you must also start with point 2 for the x-values (x₂ − x₁). Mixing order gives the wrong sign!
Example: Point 1 = (1, 2), Point 2 = (4, 8)
8 − 2 = 6
4 − 1 = 3
m = 6 ÷ 3 = 2
A vertical line has no slope (undefined) because Δx = 0, and dividing by zero is impossible. Don't write "infinite" — write undefined!
| Type | m value | Direction | Example |
|---|---|---|---|
| Positive | m = 2 | Goes up ↗ | rise=4, run=2 |
| Negative | m = −3 | Goes down ↘ | rise=−3, run=1 |
| Zero | m = 0 | Flat → | rise=0, run=any |
| Undefined | undefined | Vertical ↕ | run=0 ! |
Point 1: (x₁, y₁) = (1, 2) | Point 2: (x₂, y₂) = (4, 8)
Δy = y₂ − y₁ = 8 − 2 = 6
Δx = x₂ − x₁ = 4 − 1 = 3
m = 6 ÷ 3 = 2
m = 2 means for every 1 unit you move right, the line rises 2 units. Positive slope — the line goes up!
Slope = rise ÷ run = 6 ÷ 3 = 2. You can also verify by swapping the points: m = (2−8)÷(1−4) = −6÷−3 = 2. Same answer! ✅
Make x₁ = x₂ → watch slope become undefined (vertical line)
Make y₁ = y₂ → watch slope become 0 (flat line)
Make y₂ > y₁ → positive slope ↗ | Make y₂ < y₁ → negative slope ↘
💡 Memory Trick — Rise Over Run
Always forget which goes on top — rise or run?
Rise (Δy) goes on top. Run (Δx) goes on the bottom. Just like climbing stairs!
Assign (x₁, y₁) and (x₂, y₂) before calculating. This prevents sign errors — the most common mistake!
If both points have the same x-value (like (3, 1) and (3, 5)), the slope is undefined — never write 0!
If both points have the same y-value (like (1, 4) and (5, 4)), the slope is 0 — never write undefined!
m = 1/2 means for every 2 units right, the line rises 1 unit. GED answers often include fraction slopes — enter them as 1/2 not 0.5.
If rise = −4 and run = −2, m = −4 ÷ −2 = +2. Slope is positive — don't make sign errors!
| Situation | Answer | Example |
|---|---|---|
| Line goes up ↗ | Positive slope | m = 2 |
| Line goes down ↘ | Negative slope | m = −3/2 |
| Horizontal line → | Zero slope | m = 0 |
| Vertical line ↕ | Undefined | no slope |
| Formula | m = (y₂−y₁) ÷ (x₂−x₁) | |