By plugging $p=1-q$, into the $3$ equations show that $x=y=z$

Question

By plugging $p=1-q$, into the 3 equations: $$\begin{cases} z=py+qx \\ x=pz+qy \\ y=px+qz \end{cases}$$ show that $\boxed{x=y=z}$

This is from the final part of question 7 in this STEP paper,

and is following the advice of another students solution , only i cannot get to the required result despite the advice.

Any one able to get to $\boxed{x=y=z}$ by substituting $p=1-q$?

Kind regards,

Answer 1

The equations are equivalent to:

$$\begin{cases} z=q(x-y)+y \\ x=q(y-z)+z \\ y=q(z-x)+x \end{cases}$$

Substituting $z$ on the third equation we get:

$$y = -q^2(y-x) + q(y-x) + x \Rightarrow (y-x)(q^2-q+1) = 0$$

Similarly, $(x-z)(q^2-q+1) = 0$ and $(z-y)(q^2-q+1) = 0$.

So either $x = y = z$ or $q^2 -q +1 = 0$, but there's no real number $q$ that satisfies that equation. Therefore:

$$\boxed{x = y = z}$$