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CAMBRIDGE O LEVEL · 2025–2027

Parallel lines.

Keep the direction and change the position to pass through the required point.

Syllabus 3.6Visual notesWorked examples
BY THE END

Find parallel gradients · obtain parallel equations · handle vertical lines

01

Parallel lines have the same direction

Parallel straight lines never meet in the plane and have the same direction. On a coordinate grid, that means they make the same vertical change for every equal horizontal change.

Their gradients are equal. Their intercepts can differ, which moves one line away from the other without changing its direction. For equations not already in y = mx + c form, rearrange before comparing gradients; the raw coefficient of x may be misleading.

Distinct non-vertical parallel lines have the same gradient and different y-intercepts. Their separation stays constant; they never meet.

-6-6-5-5-4-4-3-3-2-2-1-11122334455660xy
Two distinct parallel lines have the same gradient, 1, and different y-intercepts.
y = x + 2y = x − 2
Pause and explainCan y = 2x + 1 and y = 2x − 5 be parallel?

Yes. Both have gradient 2, with different intercepts locating them at different heights.

02

Keep the gradient; find the new intercept

To find a line parallel to a known one, borrow its gradient, not its whole equation. The new line usually needs a different intercept because it must pass through a different point.

Write the new equation with the shared gradient and an unknown intercept. Substitute the required point to find that intercept. Finish by checking both conditions separately: same gradient and membership of the given point.

WORKED EXAMPLE

Find the line parallel to y = 4x − 1 through (1, −3).

  1. The given gradient is 4, so the required line is y = 4x + c.
  2. Substitute (1, −3): −3 = 4 + c.
  3. c = −7, giving y = 4x − 7.

Check the gradient is 4 and the new line contains (1, −3).

WORKED EXAMPLE

Find the line parallel to 2x + 3y = 6 through (3, 1).

The given line has gradient −23. A parallel line can also be written 2x + 3y = k.

k = 2(3) + 3(1) = 9
2x + 3y = 9
Pause and explainWhat stays the same when finding a parallel line through a new point?

The gradient. The intercept is recalculated using the new point.

03

Your turn. Match the direction.

Rearrange the original equation until its gradient is clear. Use that gradient in the new line and then substitute the stated point.

Check that you have written an equation for the new line, not merely repeated the original equation. Vertical parallel lines have equations x = different constants.

1

Find the line parallel to y = −3x + 2 through (2, 1).

Reveal worked answer
y = −3x + c. 1 = −6 + c, so c = 7. Equation: y = −3x + 7.
2

Find the line parallel to 4x − 2y = 9 through (−1, 5).

Reveal worked answer
The gradient is 2. y = 2x + c; 5 = −2 + c, so y = 2x + 7.
3

Find the line parallel to x = −2 through (5, 3).

Reveal worked answer
The required line is vertical: x = 5.