Distinct non-vertical lines with equal gradients and different intercepts are parallel. Vertical lines are parallel to one another but have undefined gradient.
y = 3x + 1 and y = 3x - 4 are parallel (both gradient 3).
To find the line parallel to y = 2x + 5 through (1, 9): use m = 2, then 9 = 2(1) + c gives c = 7, so y = 2x + 7.
Same gradient, different intercepts
Perpendicular lines
For two non-vertical lines meeting at a right angle, m₁m₂ = −1. Horizontal and vertical lines are also perpendicular, but the product rule cannot use an undefined gradient.
The perpendicular gradient is the negative reciprocal: flip the fraction and change the sign.
A line with gradient 2 has a perpendicular gradient -\frac{1}{2}; gradient -\frac{3}{4} gives \frac{4}{3}.
Negative reciprocal gradients
Rearranging first
Always rearrange to y = mx + c before reading the gradient.
2y = 6x + 4 becomes y = 3x + 2, so m = 3. x + 2y = 8 becomes y = -\frac{1}{2}x + 4, so m = -\frac{1}{2}.
Finding the equation of a parallel or perpendicular line
Step 1: find the required gradient. Step 2: substitute the given point into y = mx + c to find c.
The line perpendicular to y = 2x + 1 through (4, 3): m = -\frac{1}{2}; 3 = -\frac{1}{2}(4) + c gives c = 5, so y = -\frac{1}{2}x + 5.
Special cases
Horizontal lines (y = k) are perpendicular to vertical lines (x = k); their gradients are 0 and undefined.
Do not take the reciprocal of zero: a line perpendicular to a horizontal line is vertical, with equation x = k.
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