In the formula $Ax+By=C$, is it true that $A$ and $B$ can't both be zero? If so, why not?

Question

I read in a math book that in the formula $Ax+By=C$, I read that $A$ and $B$ can't both be zero. I think C will also be zero because anything times zero equals zero and on a graph, the x- and y- intercepts will both be zero meaning the two points will be at ($0$, $0$), so then we wouldn't be able to draw a line for our graph, meaning it would be undefined. Is this why the two mentioned variables can't both be zero?

Answer 1

If I am interpreting your question correctly, you are asking why $Ax+By=C$ is not the equation of a line when both $A$ and $B$ are zero. There are two possibilities here:

• In the first case $A=B=0$, but $C\neq 0$. Then there are no pairs $(x,y)$ which satisfy the equation, since the left hand side will be zero no matter what you plug in. Thus this equation produces no graph at all.

• In the second case, $A=B=C=0$. Here every choice of $(x,y)$ will satisfy the equation since no matter what you plug in, both sides of the equation are zero. In this case every point in the plane is on the graph, so it certainly does not produce a single line.

Answer 2

Assuming $A,B$ and $C$ are real numbers and you are trying to define a line in $\mathbb{R}^2$ then is true, $A$ and $B$ can't be both zero as the equation won't have sense unless $C$ is zero and you wouldn't be able to draw a graph of it as you said.

Now, to go a step further in the explanation. For you to be able to draw a line (or a curve or whatever) in a graph you need at least a function that depends on a variable, usually expressed as $$y=f(x)$$

Or

$$x=f(y)$$

Then for every value of $x$ (or $y$ in the second one) you can draw a point $(x,y)$ in your graph.

Notice that if both $A$ and $B$ are zero, then there is no way you could make such and expression.