Matrices and Vectors finding constant k

Question

How to work this problem.

Let Π1, Π2 and Π3 be the planes with Cartesian equations + 2 + 3 = 5, − + 2 = 7 and 3 − 3 + 9 = 10 respectively, where is a constant. (i) Find given that Π1, Π2 and Π3 do not have a unique point of intersection.

Answer 1

With $$x=5-3z-2y$$ we can eliminate $x$, we get $$13y+6z=-5$$ and $$z(2-3k)-y(2k+1)=7-15k$$ now with $$z=-\frac{5}{6}-\frac{13}{6}y$$ we get $$(-\frac{5}{6}-\frac{13}{6}y)(2-3k)-y(2k+1)=7-13k$$ now consider this equation and solve it for $y$ depending on $k$