Question

Find the vector equation for the line of intersection of the planes 5x+3y−2z=−45x+3y−2z=−4 and 5x+5z=0

Answer 1

$5x+5z=0\implies x+z=0\implies z=-x$

$5x+3y-2z=7x+3y-2x-2z=7x+3y-2=-4\implies7x+3y=-2$

Letting $x=t$ and $z=-t$, we have

$7t+3y=-2\implies y=-\dfrac73t-\dfrac23$

so the intersection is given by

$\mathbf r(t)=\left(t,-\dfrac73t-\dfrac23,-t\right)$