# Graphing cylinders and determining missing variable value.

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

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.
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