General equation of line

Question

Why can't the general equation of line (Ax+By+C=0)take both x and y values as 0, although practically a line might pass through 0,0?

Answer 1

The equation of a line is typically written as $y=mx+b$. This equation can take both $x$ and $y$ values as $0$. For example, consider the line $y=\frac{1}{3}x$ (i.e. taking $m=\frac{1}{3}$ and $b=0$). Clearly, $x=y=0$ is possible. However, do note that an equation can take both $x$ and $y$ values as $0$ if and only if $b=0$.

If you're asking why $x$ and $y$ can't both be $0$ when the line is written in the formula $ax+by+c=0$, $x$ and $y$ can both be $0$ if and only if $c=0$. This is because if $x=y=0$, then you have $c=0$, and if you have $c=0$, then $x=y=0$ will be possible in the original equation.