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 -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 VillaVilla
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