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Deriving the Point-Slope Form from the Slope Formula
The point-slope form of a line's equation can be derived directly from the slope formula. Starting with a line of slope that passes through a known point and any other point (x, y), the slope formula gives: To eliminate the fraction, multiply both sides by : The right side simplifies because the factors cancel: Finally, rewriting with the terms on the left yields the point-slope form: This derivation shows that the point-slope form is simply a rearrangement of the slope formula — the same relationship between slope, rise, and run, rewritten so that it can be used as an equation of the line rather than just a calculation of slope.
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