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 $b$.

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.

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answered Sep 22 “18 at 1:36

VillaVilla

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Given any three numbers $a$, $b$, $cin{mslsec.combb R}$ it is permitted to consider the set$$S:=igl{(x,y)in{mslsec.combb R}^2igm| ax+by=cigr}subset{mslsec.combb R}^2 .$$How this set looks like depends on the given numbers $a$, $b$, $cin{mslsec.combb R}$. One has to distinguish several cases:

(i) $ underline{a=b=c=0},:>$ Every point $(x,y)$ fulfills the condition $0x+0y=0$, hence $S={mslsec.combb R}^2$.

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(ii) $ underline{a=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.

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