<s>[INST] By plugging $p=1-q$, into the $3$ equations show that $x=y=z$ 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,
 [/INST] 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:
$$\boxed{x = y = z}$$
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