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If you have two equations for cylinders:

$$x^2 + y^2 =1$$

$$y^2 + z^2 =1$$

And you have to graph them. Th missing variables for each equation is $z$ and $x$, respectively. The cylinder will have this axis of symmetry. My book selected $z = k$ and $x = 0$ for the values for the free variables. Why is that? What is the difference? Why not just use $x=k$ or $z=0?$

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It's unclear what you mean by "selected $z=k$ and $x=0$ for the values for the free variables". What does your book do next? –  icurays1 Jan 27 '13 at 5:38
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I'm just going to assume that you're being asked to graph the surfaces defined by those equations; it sounds like your book might be a little confusing.

For both equations, we're looking a set of points $(x,y,z)$ in 3D Euclidean space. The first equation doesn't depend on $z$, which means that the cross sections parallel to the $xy$ plane will all be exactly the same. So, you can pick any cross section parallel to the $xy$ plane, figure out the 2D plot, then "extrude" this plot along the $z$ axis. For instance, you could set $z=0$ (which is fine, since your equation doesn't depend on $z$); the resulting equation describes a circle in the $xy$ plane. Do the same for $z=1$, or $z=\pi$ or in fact $z=k$ for any value of $k$ - and they will all look like circles lying in the $z=k$ plane, centered on the $z$ axis.

For the second equation, the story is much the same - your equation doesn't depend on $x$, so the cross-sections parallel to the $yz$ plane will all look the same. Thus we pick any constant value for $x$ (say, $x=0$, $x=1$, $x=k$, etc), plot the figure, then extrude along the $x$-axis.

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Do you think by free variables he means the axes of symmetry? Because in that regard, $x=0$ makes sense, $z=k$ sort of makes sense. –  Rustyn Jan 27 '13 at 5:54
    
Well, it just so happens that the "free variables" (i.e. the variables absent from the equation) are also axes of rotational symmetry, but that won't in general be the case. Really not sure what his book is getting at. –  icurays1 Jan 27 '13 at 6:02
    
Ah, I understand your interpretation. –  Rustyn Jan 27 '13 at 6:05
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