Writing the equation of a line

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Worksheet with answersDownload PDF

What this lesson covers

  • The form y − y₁ = m(x − x₁) needs one point and a gradient.
  • Two points give the gradient first.
  • Rearranging to y = mx + c shows the gradient and the y-intercept.
Key idea

The point and gradient form

  • For a line through (x₁, y₁) with gradient m, write y − y₁ = m(x − x₁).
  • Expand and tidy to reach y = mx + c, where m is the gradient and c is the y-intercept.
Worked example

Gradient 3 through (2, 5)

  • Put the numbers straight into the form.
  1. y − 5 = 3(x − 2)
  2. y − 5 = 3x − 6expand
  3. y = 3x − 1add 5
Worked example

Through (1, 4) and (3, 10)

  • Two points give the gradient, then either point finishes it.
  1. m = (10 − 4) ÷ (3 − 1) = 3
  2. y − 4 = 3(x − 1)
  3. y = 3x + 1
Worked example

Reading a line in general form

  • Rearrange 2x + 3y = 12 to see its gradient.
  1. 3y = −2x + 12
  2. y = −(2/3)x + 4gradient −2/3, y-intercept 4

Common mistakes

  • Reading the gradient of 2x + 3y = 12 as 2.
    Make y the subject first; the gradient is −2/3.
  • Forgetting to multiply the negative in y − 5 = −2(x − 3).
    The bracket expands to −2x + 6.

Your turn

  • 1. Find the line with gradient −2 through (3, 1).
  • 2. Find the line through (0, −2) and (4, 6).
Show answer

1. y = −2x + 7 2. m = 2, so y = 2x − 2

Summary

What to take away

  • One point and a gradient fix a line.
  • Make y the subject before reading the gradient.

More in AS 91256

Distance, midpoint and gradientNotesWriting the equation of a lineNotesParallel and perpendicular linesNotesModelling problems on the coordinate planeNotes

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