Consider the equation $ax+by=c$ in 2-space and the slope for that equation where $a$ and $b$ are real numbers.

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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 $b$.


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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:

$$y=frac cb -fracabx$$

So the slope is $fracdydx=frac-ab$. 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.


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answered Sep 22 "18 at 1:36
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Given any three numbers $a$, $b$, $cinmmsanotherstage2019.combb R$ it is permitted to consider the set$$S:=igl(x,y)inmmsanotherstage2019.combb R^2igmsubsetmmsanotherstage2019.combb R^2 .$$How this set looks like depends on the given numbers $a$, $b$, $cinmmsanotherstage2019.combb R$. One has to distinguish several cases:

(i) $ underlinea=b=c=0,:>$ Every point $(x,y)$ fulfills the condition $0x+0y=0$, hence $S=mmsanotherstage2019.combb R^2$.

(ii) $ underlinea=b=0 wedge c e0,:>$ No point $(x,y)$ fulfills the condition $0x+0y=c$, hence $S=emptyset$ (the empty set).

(iii) $ underline(a,b) e(0,0),: $ The set $S$ is a line. If $b=0$ (hence $a e0$) then the condition $ax+by=c$ is equivalent with $x=-cover a$, and the line $S$ is vertical. If $a=0$ (hence $b e0$) then the line $S$ is given by $y=-cover b$, and is horizontal.

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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.