Equations of straight lines

Gradient

The gradient through (x_1, y_1) and (x_2, y_2) is m = (y_2 - y_1) / (x_2 - x_1).

Three forms

• y = mx + c: gradient m, y-intercept c
• y - y_1 = m(x - x_1): through (x_1, y_1) with gradient m
• ax + by + c = 0: often asked for with a, b and c integers

Intercepts and intersections

Put x = 0 for the y-intercept and y = 0 for the x-intercept. Two lines meet where both equations are true, so solve them simultaneously.

Worked example

Find the equation of the line through (2, 3) and (6, 11) in the form ax + by + c = 0.

1. m = (11 - 3)/(6 - 2) = 2
2. y - 3 = 2(x - 2), so y = 2x - 1
3. Rearrange: 2x - y - 1 = 0

Answer: 2x - y - 1 = 0

Check yourself

1. Find the gradient of the line through (-1, 4) and (3, -8).
   Show answer

   -3. m = (-8 - 4)/(3 + 1) = -12/4 = -3.

2. A line has gradient 4 and passes through (2, 5). Find its y-intercept.
   Show answer

   -3. y = 4x + c, and 5 = 8 + c, so c = -3.

3. Find the gradient of the line 3x + 2y - 6 = 0.
   Show answer

   -1.5. 2y = -3x + 6, so y = -1.5x + 3. The gradient is -1.5.

4. Where does 3x + 2y - 6 = 0 cross the x-axis? Give the x-coordinate.
   Show answer

   2. Put y = 0: 3x = 6, so x = 2.

5. A line passes through (1, -2) with gradient 3. Which is its equation?
   • 3x - y - 5 = 0
   • 3x - y + 5 = 0
   • 3x - y - 1 = 0
   • x - 3y - 7 = 0
   Show answer

   3x - y - 5 = 0. y + 2 = 3(x - 1), so y = 3x - 5, which rearranges to 3x - y - 5 = 0.

6. Find the x-coordinate where y = 2x + 1 meets y = -x + 7.
   Show answer

   2. 2x + 1 = -x + 7, so 3x = 6 and x = 2.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 1 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.