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Imagine you are tracking an airplane's flight path: you know its current coordinate and its heading, but air traffic control systems require equations set to zero.
Different geometric problems demand different algebraic disguises for the exact same straight line.

Anatomy of Both Forms

The point-slope form of a line is y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm represents the steepness or slope, and (x1,y1)(x_1, y_1) is any known point on the line.
This comes directly from the definition of slope as the ratio of vertical change to horizontal change: m=y−y1x−x1m = \frac{y - y_1}{x - x_1}, multiplied across by (x−x1)(x - x_1).
The general form is written as Ax+By+C=0Ax + By + C = 0, where AA, BB, and CC are integers with no common factor, and AA is conventionally non-negative (A≥0A \ge 0).
Why do mathematicians bother rewriting point-slope form when it is already so convenient?

Point-Slope to General Form

Converting to general form requires two main actions: eliminating all fractional denominators and moving every term to one side.
Let's convert the line passing through (4,−1)(4, -1) with a slope of m=−25m = -\frac{2}{5}, which starts as y−(−1)=−25(x−4)y - (-1) = -\frac{2}{5}(x - 4).
First, simplify the double negative to get y+1=−25(x−4)y + 1 = -\frac{2}{5}(x - 4), then multiply both sides by the denominator 55 to get 5(y+1)=−2(x−4)5(y + 1) = -2(x - 4).
Expanding both sides yields 5y+5=−2x+85y + 5 = -2x + 8.