Straight Lines

Because rewriting $ax+by+c=0$ (dividing throughout by whichever of $a,b$ is nonzero) always removes one constant, a straight line's equation genuinely contains only two independent arbitrary constants — so two pieces of information about the line (two points on it, or a point and its slope, or its two intercepts, etc.) are always exactly enough to determine it uniquely, and never more than enough. The different named 'forms' of a line's equation below are simply the same two-constant equation, organised around whichever pair of data happens to be given in a problem: (i) slope and intercept, (ii) a point and the slope, (iii) two points, (iv) the two intercepts, plus two special forms useful in particular settings — (v) the nor …