Find the equation of the plane that passes through the point $(1,-2,3)$ and is parallel to the plane $5x-y+2z=7$.

Step-by-step solution

1. Recognise that parallel planes have the same normal vector; thus the required plane has normal $\mathbf{n}=⟨5,-1,2⟩$.
2. Write the general equation of a plane with this normal as $5x-y+2z=D$, where $D$ is a constant to be found.
3. Substitute the coordinates of the given point $(1,-2,3)$ into the equation:
4. Compute $5(1)-(-2)+2(3)=5+2+6=13$, so $D=13$.
5. Hence the required plane is $5x-y+2z=13$.

Concept tested: Plane parallel to a given plane

Published 14 September 2026 · How this solution was prepared and checked