Given sphere with equation $x^2+y^2+z^2=100$, and plane $x=5$, describe the intersection of these two surfaces.
I understand that I could plugin $5$ for $x$. However, this yields a cylinder that does not describe the specific circle. How do I write the equation of this circle?
The graph in $\mathbb{R}^3$ of a single equation in the variables $x,y,z$ is typically a surface, not a curve.
The circle in question can be represented algebraically using the system of two equations $$ y^2+z^2=75,\;\;\;x=5 $$ or alternatively, in parametric form \begin{align*} x&=5\\[4pt] y&=5\sqrt{3}\cos(t)\\[4pt] z&=5\sqrt{3}\sin(t)\\[4pt] \end{align*} for $0\le t < 2\pi$.
It's possible to represent the circle in question as the graph in $\mathbb{R}^3$ of the single equation $$ (x-5)^2+(y^2+z^2-75)^2=0 $$ but such a representation is not very useful.