I need to find the parametrization of the intersection for the two surfaces defined by these equations: $3x-4y+5z=3$ and $-x+2y-4z=10$ I'm not really sure how to do it. What is the best way?
2026-04-03 01:50:27.1775181027
Parametrization of the intersection of two linear surfaces
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$\textbf{Hint}$: Set $z = 0$ (or any constant you like), then solve the remaining system of two variables.
$\textbf{Comment}$: Honestly, you have a choice in which variable you want to eliminate in the initial step, I usually just go for $z$. In either case you will left to solve a system of 2 linear equations with 2 unknowns.
The above will give you a point $p$ in the intersection of the two planes. The parametrization of the line of intersection is $p+ t(\vec{n_1} \times \vec{n}_2)$ where $\vec{n}_i$ is the normal for plane $P_i$.