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=-\frac73t-\frac23`

so the intersection is given by

`\mathbf r(t)=\left(t,-\frac73t-\frac23,-t\right)`