Step by Step
1
Identify rise (vertical change)
Rise is the change in the y-values between two points: y₂ − y₁.
2
Identify run (horizontal change)
Run is the change in the x-values between the same two points: x₂ − x₁.
3
Divide rise by run
Slope = (y₂ − y₁) / (x₂ − x₁). Keep the point order consistent in both the numerator and denominator — don't swap which point is "point 1" halfway through.
4
Read the sign
A positive slope rises left to right; a negative slope falls left to right.
Applied Walkthrough
1
Find the slope between points (1, 2) and (4, 8). Rise: y₂ − y₁ = 8 − 2 = 6.
2
Run: x₂ − x₁ = 4 − 1 = 3.
3
Slope = 6/3 = 2. Since the slope is positive, the line rises as you move left to right.
4
The mnemonic "you rise UP before you run ACROSS" keeps the numerator (rise/vertical) and denominator (run/horizontal) in the right place.
Exam Application
Exams test whether you can correctly calculate rise over run from two coordinate points, keep point order consistent between numerator and denominator, and correctly interpret the sign of the result.
⚠ Common Trap
The most common trap is flipping the formula — calculating run over rise by accident, or mismatching which point is "first" in the numerator versus the denominator. Keep the same point order in both.
✓ Quick Self-Check
1. What is the slope formula?
(y₂ − y₁) / (x₂ − x₁).
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2. Find the slope between (1,2) and (4,8).
2 (rise of 6 over run of 3).
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3. What does a negative slope indicate about a line?
The line falls as you move left to right.
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4. What is "rise" in the slope formula?
The vertical change between two points, y₂ − y₁.
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5. What is a common error when calculating slope?
Flipping the formula (calculating run over rise) or mismatching point order between numerator and denominator.
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