Equation of a Line

310. The Point-Slope Promise · how a fixed point and a slope define a line

The ratio of vertical change to horizontal change between P and Q always equals the slope m.

The point–slope form writes a line through a known point (x₁, y₁) with slope m as y − y₁ = m(x − x₁). It holds because the slope between (x₁, y₁) and any other point (x, y) on the line equals m.

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Selina ICSE: Equation of a Line

What this lesson covers

Try to break it

Drag Q anywhere along the line. The triangle PQR has vertical leg (y − y₁) and horizontal leg (x − x₁); their ratio is always m. Rearrange and you get y − y₁ = m(x − x₁). Try to find a Q on the line that violates this; impossible.

How you build it

Build the slope triangle through a fixed point and find the slope.

Point tool: plot the fixed point P(x₁, y₁) at (-2, -1).
Point tool: from P go 4 right and 3 up to Q(2, 2). Run = 4, rise = 3, so slope m = 3/4 = 0.75.
Line tool: click P, then Q — the line through the fixed point P with slope m.
Point tool: mark R at (2, -1) — directly below Q and level with P. This is the right-angle corner.
Segment tool: join P to R — the horizontal run, x − x₁ = 4.
Segment tool: join R to Q — the vertical rise, y − y₁ = 3. So slope m = (y − y₁)/(x − x₁) = 3/4 = 0.75.

The proof, step by step

Prove that the line through P with slope m satisfies y − y₁ = m(x − x₁).

Identify P as (x₁, y₁) and Q as (x, y).
Form right triangle PQR with horizontal leg PR and vertical leg QR.
Length QR = y - y₁ and length PR = x - x₁.
Slope m = tan θ = QR / PR = (y - y₁) / (x - x₁).
Rearrange to get y - y₁ = m(x - x₁).

Worked example

Find the equation of the line with slope 3 passing through the point (2, -1).

Substitute m=3, x₁=2, y₁=-1 into y - y₁ = m(x - x₁) to get y - (-1) = 3(x - 2), which simplifies to y + 1 = 3(x - 2).

y + 1 = 3(x - 2) — correct
y - 1 = 3(x + 2)
y + 1 = -3(x - 2)
y - 1 = -3(x + 2)