Question

Show that every line has an equation of the form \(A x+B y=C\), where \(A\) and \(B\) are not both 0 , and that, conversely, every such equation is the equation of a line.

Answer 1

If a given line is vertical, it has an equation \(x=C\). In this case, we can let \(A=1\) and \(B=0\). If the given line is not vertical, it has a slope-intercept equation \(y=m x+b\), or, equivalently, \(-m x+y=b\). So, let \(A=-m, B=1\), and \(C=b\). Conversely, assume that we are given an equation \(A x+B y=C\), with \(A\) and \(B\) not both 0 . If \(B=0\), the equation is equivalent to \(x=C / A\), which is the equation of a vertical line. If \(B \neq 0\), solve the equation for \(y: \quad y=-\frac{A}{B} x+\frac{C}{B}\). This is the slope-intercept equation of the line with slope \(-\frac{A}{B}\) and \(y\)-intercept \(\frac{C}{B}\).