The General Equation for a Straight Line

Any nonvertical line in the plane has the equation y = ax + b. A vertical line, that is parallel to the y-axis, will intersect the x-axis at some point (c, 0). Every point on the line has the same *-coordinate c, so the line must be x = c

This is the equation for a straight line through (c, 0) parallel to the y-axis.

The equations y = ax + b and x = c can both be written as

for suitable values of the constants A, B, and C. Specifically, >• = ax + b corresponds to A = a, B = — 1, and C — b. whereas x = c corresponds to A = 1, B = 0. and C = — c. Conversely, every equation of the form [2.9] represents a straight line in the plane, disregarding the uninteresting case when A = B = 0. If B = 0. it follows from [2.9] that Ax = —C, or x = —C/A. This is the equation for a straight line parallel to the y-axis. On the other hand, if B ^ 0, solving [2.9] for y yields y~ BX B

This is the equation for a straight line with slope —A/B. Equation [2.9] thus deserves to be called the general equation for a straight line in the plane.

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