Vectors and Matrices how to solve

Question

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 k = 2/3

Answer 1

From the first equation we get $$x=5-2y-3z$$ plugging this in the second equation we obtain $$9y+18z=5$$ and from the third equation (with the $x$ above) we have $$z(2-3k)-y(1+2k)=7-5k$$ plugging $$z=\frac{5}{18}-\frac{1}{2}y$$ in the last equation above: $$(\frac{5}{18}-\frac{1}{2}y)(2-3k)-y(1+2k)=7-5k$$ simplifying we obtain $$-{\frac{67}{9}}+{\frac {17\,k}{3}}+2\,y-3\,yk-y \left( 1+2\,k \right) =0$$ Can you finish? Simplifying this we get $$(-3+k)y=\frac{67}{9}-\frac{17}{3}k$$