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MathidiaMathematics Academy
Coordinate geometry and circles
Lesson 11 of 75
Introduction

Equations of straight lines

Objective. Move fluently between point-gradient and Cartesian line forms.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A straight line can be written in point-gradient form y−y₁=m(x−x₁), or rearranged into Cartesian form ax+by+c=0 with integer coefficients and no fractions. The gradient m between two points (x₁,y₁) and (x₂,y₂) is (y₂−y₁)/(x₂−x₁). For a line through (1,2) with gradient 3, point-gradient form gives y−2=3(x−1), which expands to y=3x−1, or rearranged to 3x−y−1=0. Always check the final Cartesian form has no common factor left unsimplified and matches the original point when substituted back.

Worked example

Find the equation of the straight line passing through (2,−1) with gradient 4, in the form y=mx+c.

in the form y=mx+c
Coordinate grid with the line in the form y=mx+c
  1. 1Use point-gradient form: y−(−1)=4(x−2)
  2. 2Simplify the left side: y+1=4(x−2)
  3. 3Expand the right side: y+1=4x−8
  4. 4Rearrange to get y=4x−9

Answer: y = 4x − 9

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Practice

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Common mistake. Students compute gradient as (x₂−x₁)/(y₂−y₁), inverting the fraction; for points (0,0) and (2,4) this gives 0.5 instead of the correct gradient 2, so always keep y-difference on top.

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