Home / IGCSE Maths 0580 / Coordinate geometry / Parallel lines
Cambridge IGCSE · Topic 3.6
IGCSE Maths 0580 / Topic 3 / IGCSE

Same slope, never meet.

Parallel lines climb at exactly the same rate, so they have the same gradient. Keep the gradient, change the intercept, and you have a new parallel line through any point you like.

Full-screen labStraight lines lab → Topic 3All coordinate geometry lessons →
The key idea

Parallel lines have equal gradients. To find the line parallel to \(y = mx + c\) through a point: keep the same \(m\), substitute the point to find the new \(c\).

−1 1 2 3 4 5 6 −5 −4 −3 −2 −1 1 2 3 4 5 6 x y y = 2x + 1 y = 2x − 5 (3, 1) same gradient: 2 (rise 2 for every 1 across) different y-intercepts: 1 and −5
Fig. 1: y = 2x + 1 and y = 2x - 5 both rise 2 for every 1 across, so they are parallel; the second one passes through (3, 1)
Section 01

Same gradient, different intercept.

In the lab, drag the point that the second line must pass through. The new line always keeps the gradient of the first one, so its gradient triangle is identical; only its intercept changes. Note the equation of the new line for two or three positions of the point.

Worked example Spotting parallel lines
Which of these lines are parallel?   A: \(y = 3x + 2\)   B: \(y = 2x + 3\)   C: \(y = 3x - 5\)   D: \(y = 5 - 3x\)
  1. Gradients: A is 3, B is 2, C is 3.
  2. D is \(y = -3x + 5\), so its gradient is \(-3\), not 3.
  3. Only A and C have the same gradient.
Answer: A and C
Worked example Parallel line through a point
Find the equation of the line parallel to \(y = 4x - 1\) that passes through \((1, -3)\).
  1. Parallel, so the gradient is also 4: \(y = 4x + c\).
  2. Substitute \(x = 1\), \(y = -3\): \(-3 = 4 \times 1 + c\).
  3. \(c = -3 - 4 = -7\).
Answer: \(y = 4x - 7\)
Examiner trap

Using the point's \(y\)-coordinate as the intercept. The line through \((1, -3)\) is NOT \(y = 4x - 3\): the point is not on the \(y\)-axis, so you must substitute to find \(c\).

Section 02

Parallel to ax + by = c.

(Extended) When the given line is in a form like \(2x + 3y = 6\), rearrange to \(y = mx + c\) to read its gradient. Or use the shortcut: a parallel line keeps the same left-hand side, \(2x + 3y = k\), and only \(k\) changes.

Worked example Rearranging first (Extended)
Find the equation of the line parallel to \(2x + 3y = 6\) that passes through \((3, 1)\).
  1. \(3y = -2x + 6\), so \(y = -\dfrac{2}{3}x + 2\). The gradient is \(-\dfrac{2}{3}\).
  2. New line: \(y = -\dfrac{2}{3}x + c\). Substitute \((3, 1)\): \(1 = -2 + c\), so \(c = 3\).
  3. Shortcut check: \(2x + 3y = k\) with \(k = 2 \times 3 + 3 \times 1 = 9\). Multiplying \(y = -\dfrac{2}{3}x + 3\) by 3 gives \(3y = -2x + 9\), the same line.
Answer: \(y = -\dfrac{2}{3}x + 3\), or \(2x + 3y = 9\)
Worked example Through the origin
Find the equation of the line parallel to \(y = 5 - x\) that passes through the origin.
  1. \(y = 5 - x = -x + 5\), so the gradient is \(-1\).
  2. Through \((0, 0)\), so \(c = 0\).
Answer: \(y = -x\)
Your turn
(Extended) Find the equation of the line parallel to \(3x - y = 4\) that passes through \((-2, 5)\).
Examiner trap

Reading the coefficient of \(x\) in \(2x + 3y = 6\) as the gradient. The gradient is \(-\dfrac{2}{3}\), not 2: rearrange to \(y = \ldots\) before you read it.

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Which line is parallel to \(y = 3x - 4\)?

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What is the gradient of any line parallel to \(y = 6 - 2x\)?

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Two different parallel lines always have

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Which is the line parallel to \(y = 2x + 5\) that passes through the origin?

Section 03

Using parallel lines.

Equal gradients prove that sides are parallel, which identifies parallelograms and trapezia. You can also choose an unknown so that two lines are parallel.

Worked example Proving a parallelogram
Show that \(A(0, 1)\), \(B(4, 3)\), \(C(5, 7)\) and \(D(1, 5)\) form a parallelogram \(ABCD\).
  1. Gradient \(AB = \dfrac{3 - 1}{4 - 0} = \dfrac{1}{2}\); gradient \(DC = \dfrac{7 - 5}{5 - 1} = \dfrac{1}{2}\).
  2. Gradient \(AD = \dfrac{5 - 1}{1 - 0} = 4\); gradient \(BC = \dfrac{7 - 3}{5 - 4} = 4\).
  3. Both pairs of opposite sides are parallel.
Answer: \(AB \parallel DC\) and \(AD \parallel BC\), so \(ABCD\) is a parallelogram.
Worked example Choosing a gradient (Extended)
Find the value of \(k\) for which \(y = kx + 2\) is parallel to \(6x - 2y = 5\).
  1. \(-2y = -6x + 5\), so \(y = 3x - 2.5\). The gradient is 3.
  2. Parallel lines have equal gradients, so \(k = 3\).
Answer: \(k = 3\)
Examiner trap

Dividing by a negative and losing the sign. From \(-2y = -6x + 5\), every term is divided by \(-2\): \(y = 3x - 2.5\). Check the sign of your gradient by picturing which way the line slopes.

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Find the equation of the line parallel to \(y = 4x - 1\) that passes through \((1, -3)\).

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(Extended) What is the gradient of a line parallel to \(3x + 4y = 10\)?

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(Extended) Find the equation of the line parallel to \(2x + 3y = 6\) that passes through \((3, 1)\).

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(Extended) The line \(y = kx + 2\) is parallel to \(6x - 2y = 5\). Find \(k\).

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Parallel lines, lined up.

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Six original Cambridge-style questions
Spotting parallel lines, parallel lines through a point, rearranging \(ax + by = c\), a trapezium proof and two cyclists who never meet.