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Equations of Straight Lines

Given a point and a gradient — or two points — you can write the equation of any straight line. The point-gradient formula $y - y_{1} = m (x - x_{1})$ is the most flexible tool for the job.

25 min Video by Zeeshan Zamurred Straight Line Graphs

Edexcel Straight Lines — full chapter playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Use y − y₁ = m(x − x₁) to find a line
Find a line through two points
Convert to the form ax + by + c = 0
Find where a line crosses the axes

Point-gradient form

If you know the gradient $m$ and one point $(x_{1}, y_{1})$ , the line is given directly by the point-gradient formula.

y - y_{1} = m (x - x_{1})

Substitute the gradient and the point, then simplify.

y - 3 = 4 (x - 2)

2Expand:

y = 4 x - 5

Answer

y = 4 x - 5

Through two points

With two points, first find the gradient, then use either point in the point-gradient formula.

1Gradient

= \frac{7 - 1}{3 - 1} = 3

2Use

(1, 1)

y - 1 = 3 (x - 1)

Answer

y = 3 x - 2

Tip — Always find the gradient first when given two points, then plug into y − y₁ = m(x − x₁).

The form ax + by + c = 0

Exam answers are often required in the form $a x + b y + c = 0$ with integer coefficients. Rearrange and multiply through to clear fractions.

1Multiply by 2:

2 y = x + 6

2Rearrange:

x - 2 y + 6 = 0

Answer

x - 2 y + 6 = 0

Formula recap

y - y_{1} = m (x - x_{1})

Point-gradient form.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Gradient from two points.

a x + b y + c = 0

Standard exam form.

Common mistakes to avoid

Forgetting to find the gradient before using the formula.

With two points, compute m first.

Leaving fractions in ax + by + c = 0.

Multiply through to clear fractions and use integer coefficients.

Key takeaways

Point-gradient form: y − y₁ = m(x − x₁).
For two points, find the gradient first.
Rearrange to ax + by + c = 0 with integers when asked.

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