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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.
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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 $\frac{C}{B}$.
by Diamond (88,237 points)

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