Ad Banner Placeholder

Cambridge IGCSE Mathematics — 0580 Extended

Topic 3.6: Coordinate geometry — Parallel Lines

Equal gradients

Two straight lines are parallel if and only if they have the same gradient (and are not the same line). They never meet, however far you extend them.

Show that \(y = 2x + 1\) and \(y = 2x - 3\) are parallel.

Two parallel lines both with gradient 2: y equals 2x plus 1 and y equals 2x minus 3
Both have \(m = 2\), so they are parallel. Different \(c\) means they are distinct lines.

Method

  1. Read (or rearrange to find) the gradient of the given line.
  2. A parallel line has exactly the same \(m\).
  3. Use the given point to find \(c\) (or use \(y - y_1 = m(x - x_1)\)).

Parallel line through a point

Keep the gradient. Change the intercept so the line passes through the given point.

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

Steps: keep gradient 4, substitute point (1, negative 3) into point-gradient form to get y equals 4x minus 7
\(m = 4\); \(y + 3 = 4(x - 1)\) simplifies to \(y = 4x - 7\).

Paper 2 (non-calculator)

Check by substituting the point back into your answer: for \(y = 4x - 7\), when \(x = 1\), \(y = -3\). If the point does not satisfy the equation, the intercept is wrong.

Given line in \(ax + by = c\) form

If the given line is not already \(y = mx + c\), rearrange first to find \(m\). Then proceed exactly as before.

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

Steps: rearrange 2x plus y equals 6 to find m equals negative 2, then get y equals negative 2x plus 7
\(y = -2x + 6\) so \(m = -2\); through \((3, 1)\): \(y = -2x + 7\).

Try this

Find the equation of the line parallel to \(3x - y = 4\) that passes through \((-1, 2)\).

Show answer
Answer

  1. Rearrange to find \(m\)

    \[ -y = -3x + 4 \Rightarrow y = 3x - 4 \Rightarrow m = 3 \]

  2. Same \(m\), through \((-1, 2)\)

    \[ y - 2 = 3\bigl(x - (-1)\bigr) = 3(x + 1) \]

  3. Fully simplified

    \[ y = 3x + 5 \]

0/10

Ad Banner Placeholder