Algebra · Straight-line graphs
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Parallel and perpendicular lines
Why parallel lines share a gradient (and nothing else), why perpendicular gradients multiply to −1 (the negative reciprocal), and how to build a line through a given point with either condition.
Algebra · Straight-line graphs
Parallel and perpendicular lines
Why parallel lines share a gradient (and nothing else), why perpendicular gradients multiply to −1 (the negative reciprocal), and how to build a line through a given point with either condition.
Why it works
Parallel = same steepness = same gradient. Two lines that never meet climb at the same rate: and are parallel — same , different . The intercept is irrelevant to parallelism; it just slides the line up or down. So "parallel to through " is instant: keep , read the new intercept: .Perpendicular = negative reciprocal. Turning a line through 90° swaps its rise and run AND flips one sign: a gradient of (right 1, up 4) becomes right 4, DOWN 1 — gradient . In general
Both moves are required: (reciprocal, no flip) and (flip, no reciprocal) are the two classic half-answers. The product test is the clean check: . ✓
"Show these lines are perpendicular" = show the product is −1. and : rearrange the second — , gradient . Product: — perpendicular, and that one line of arithmetic IS the proof. (Never read a gradient off an un-rearranged form.)
Through a given point: condition first, then the usual recipe. For the line perpendicular to through : the gradient must be , then from the point — , so :
The given point is not decoration: the perpendicular gradient alone describes a whole family of lines; the point picks the one.