[tex]5x+5z=0\implies x+z=0\implies z=-x[/tex]
[tex]5x+3y-2z=7x+3y-2x-2z=7x+3y-2=-4\implies7x+3y=-2[/tex]
Letting [tex]x=t[/tex] and [tex]z=-t[/tex], we have
[tex]7t+3y=-2\implies y=-\dfrac73t-\dfrac23[/tex]
so the intersection is given by
[tex]\mathbf r(t)=\left(t,-\dfrac73t-\dfrac23,-t\right)[/tex]
