Consider the equation $ax+by=c$ in 2-space and the slope for that equation where $a$ and $b$ are real numbers.
Explain why the slope of the equation is defined only for nonzero values of $b$. What happens when $b$ is zero?
When $b=0$ the equation is undefined meaning it doesn't exist. But I am having trouble with explaining why the slope of the equation is defined only for nonzero values of $b$.
On
Given any three numbers $a$, $b$, $c\in{\mathbb R}$ it is permitted to consider the set $$S:=\bigl\{(x,y)\in{\mathbb R}^2\bigm| ax+by=c\bigr\}\subset{\mathbb R}^2\ .$$ How this set looks like depends on the given numbers $a$, $b$, $c\in{\mathbb R}$. One has to distinguish several cases:
(i) $\ \underline{a=b=c=0}\,:\>$ Every point $(x,y)$ fulfills the condition $0x+0y=0$, hence $S={\mathbb R}^2$.
(ii) $\ \underline{a=b=0\ \wedge\ c\ne0}\,:\>$ No point $(x,y)$ fulfills the condition $0x+0y=c$, hence $S=\emptyset$ (the empty set).
(iii) $\ \underline{(a,b)\ne(0,0)}\,:\ $ The set $S$ is a line. If $b=0$ (hence $a\ne0$) then the condition $ax+by=c$ is equivalent with $x=-{c\over a}$, and the line $S$ is vertical. If $a=0$ (hence $b\ne0$) then the line $S$ is given by $y=-{c\over b}$, and is horizontal.
Of course you knew all this before, but for no reason had the feeling that for certain values of the parameters $a$, $b$, $c$ the equation $ax+by=c$ is "forbidden" or "doesn't exist". This is definitely not the case. However, the following is true: If $b=0$ then the line $S$ has no slope.
On
When $b=0$ we get $x=\operatorname{const}$. So the $x$-coordinate is fixed, and $y$ is free to be anything at all. This describes a vertical line. Vertical lines have infinite, or undefined, slope. If you take two points on a vertical line and try computing the slope, then since the slope $m=\frac{\Delta y}{\Delta x}$, we get $0$ in the denominator (the two $x$-coordinates are the same).
The slope is zero when it is horizontal, and it approaches infinity when it gets vertical. See that when you have $b=0$, your line is a vertical one in $x=\frac ca$. You can´t have a slope with value $\infty$, so it is undefined. The equation can be written as:
$$y=\frac cb -\frac{a}{b}x$$
So the slope is $\frac{dy}{dx}=\frac{-a}{b}$. There you can see that for $b=0$ the slope is undefined. Also, see that when $b=0$, the $y$ term vanishes, so it isn't even a function. That is not the case if $a=0$, because then you have $y=\frac cb$, and then $y$ (the value of the function) is perfectly defined as $\frac cb$ for all $x$, and the slope is obviously $0$, because the function becomes a horizontal line.