What makes lines parallel
Two lines are parallel when they point in the same direction and so never cross, no matter how far you extend them. Because the gradient is what fixes a line's direction, parallel lines share the same gradient.
- Parallel lines
- Lines with equal gradients but different \(y\)-intercepts. Equal gradients keep them the same steepness, and different intercepts keep them apart.
Spotting parallel lines
When both equations are in the form \(y = mx + c\), just compare the number in front of \(x\). For example, \(y = 2x + 1\) and \(y = 2x - 7\) are parallel, since both have gradient \(2\). If an equation is not in that form yet, rearrange it first, then compare gradients.
Same gradient, different intercept
If two equations share the gradient and the intercept, they are the same line, not a genuine pair of parallel lines. Parallel lines need matching \(m\) but different \(c\).
Finding a parallel equation
To find a line parallel to a given one through a chosen point: copy the gradient, then substitute the point to find the new intercept \(c\).
Worked example
Find the equation of the line parallel to \(y = 3x - 4\) that passes through \((1, 5)\).
Check yourself
1. Are \(y = 2x + 1\) and \(y = 2x - 7\) parallel? +
Both have gradient \(2\) and different intercepts, so yes, they are parallel.
2. What gradient does any line parallel to \(y = -5x + 3\) have? +
Parallel lines copy the gradient, so it must be \(\mathbf{-5}\).
3. Find the line parallel to \(y = 4x + 1\) through \((0, -3)\). +
Keep \(m = 4\). The point \((0, -3)\) is the \(y\)-intercept, so \(c = -3\). The equation is \(\mathbf{y = 4x - 3}\).
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