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Coordinate geometry

Point-slope form

The equation of a line through a known point with a known slope, without first finding the intercept.

y−y1=m(x−x1)

Calculator

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See it move

−12−10−8−6−4−224681012−6−4−2246run 2rise −1(2, 3)
  • y − 3 = −0.5(x − 2)
  • Same line: y = −0.5x + 4

What the symbols mean

y
the y-coordinate of any point on the line
x
the x-coordinate of that point
m
the slope
x1
the x-coordinate of the known point
y1
the y-coordinate of the known point

When to use it

Use it when you know one point on a line and its slope: the tangent to a curve at a point, a line through a given point parallel or perpendicular to another, or a line through two points (find the slope first, then use either point). Expand and rearrange to get slope-intercept form.

Worked example

A line passes through (2, 3) with slope m = −0.5. Find y on the line when x = 6.

  1. Start from the formula.

    y−y1=m⁢(x−x1)

  2. Rearrange to make y the subject: add y₁ to both sides.

    y=y1+m⁢(x−x1)

  3. Substitute x = 6, m = −0.5, x1 = 2, y1 = 3 and work it out.

    y=3+(−0.5·(6−2))
    =3+(−0.5·4)
    =3+(−2)
    =1

  4. In slope-intercept form the whole line is y=−0.5x+4, because its y-intercept is y1−mx1.

Answer y = 1

Why it works

Any other point (x, y) on the line, together with (x1,y1), must give the line's slope: y−y1x−x1=m. Multiply both sides by x−x1.

Common mistakes

  • Sign errors with negative coordinates: through (−2, 5) the form is y−5=m(x+2).
  • Swapping x and y: the y-coordinates go on the left, the x-coordinates inside the bracket.
  • Forgetting to multiply m into both terms of the bracket when expanding.