Equation of a Line

305. Rise Over Run · The secret formula for a line's steepness

The slope calculated from coordinates always matches rise/run.

O-9-8-7-6-5-4-3-2-1123456789-6-5-4-3-2-1123456RP = (-4, -2)P = (-4, -2)Q = (4, 2)Q = (4, 2)rise = y₂−y₁ = 4rise = y₂−y₁ = 4run = x₂−x₁ = 8run = x₂−x₁ = 8m = (y₂−y₁)/(x₂−x₁) = 0.5m = (y₂−y₁)/(x₂−x₁) = 0.5PQ
The slope of a line measures steepness as rise over run — the change in y divided by the change in x: m = (y₂ − y₁)/(x₂ − x₁). A positive slope rises to the right, a negative slope falls.

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Selina ICSE: Equation of a Line

What this lesson covers

Try to break it

Drag P and Q around. The slope readout always equals (y₂ − y₁) / (x₂ − x₁) — rise over run. Align them vertically (same x) and the denominator hits 0; the line goes vertical and the slope blows up to "undefined". Align them horizontally (same y) and the slope reads 0 — a flat line.

How you build it

Build the slope triangle of a line and find its slope.

  • Point tool: plot P at (-4, -2) on the grid.
  • Point tool: plot Q at (4, 2).
  • Segment tool: join P to Q — the line whose slope we want.
  • Point tool: mark R at (4, -2) — directly below Q and level with P. This is the right-angle corner.
  • Segment tool: join P to R — the run, the horizontal change x₂−x₁ = 8.
  • Segment tool: join R to Q — the rise, the vertical change y₂−y₁ = 4. Slope m = rise / run = 4 / 8 = 0.5.

The proof, step by step

Prove that the slope between two points equals rise over run.

  • Identify the coordinates of P as (x₁, y₁) and Q as (x₂, y₂).
  • The horizontal distance (run) between P and R is x₂ – x₁.
  • The vertical distance (rise) between R and Q is y₂ – y₁.
  • Slope m is defined as rise divided by run.
  • Therefore, m = (y₂ – y₁) / (x₂ – x₁).

Worked example

A straight line passes through the points A(2, 3) and B(6, 11). Calculate the slope of the line.

Using the slope formula m = (y₂ – y₁)/(x₂ – x₁), substitute the coordinates: m = (11 – 3)/(6 – 2) = 8/4 = 2.

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  • 2 — correct
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