Take $x$, $s$ as variables and $v,w,y$ as coefficients. Consider the augmented coefficient matrix: $$(A|b)=\left(\begin{array}{cc|c} w & y & v\\ w^2 & y^2 & v^2\\ w^3 & y^3 & v^3\\ \end{array}\right) \sim \left(\begin{array}{cc|c} w & y & v \\ 0 & y^2-wy & v^2-wv \\ 0 & 0 & v^3-wv^2-v^2y+wvy\\ \end{array}\right)$$Now we use following Theorem: Kronecker-Capelli's Theorem: Let $A$ be a coefficient matrix and $(A|b)$ be an augmented coefficient matrix w.r.t a system of equations. Then this system of equations has (at least) one solution if and only if $\text{rank}(A) = \text{rank}(A|b)$. Since $v,w,x,y,s$ should satisfy the above system of equations, we must have $2=\text{rank}(A) = \text{rank}(A|b)$, so $v^3-wv^2-v^2y+wvy = v(v^2-(w+y)v+wy)=0$. Case 1: $v=0$ Then $wx=-ys$ and thus $0=w^2x-wxy=wx(w-y)$. Either $w$ or $x$ is $0$ (which would give $v=w$ or $v=x$) or $w-y= 0$ (which would give $w=y$). Case 2: $v^2-(w+y)v+wy=0$. Solving this quadratic equation in $v$ gives either $v=w$ or $v=y$. Therefore, there exist (at least) two of $v,w,x,y,s$ that are equal. $\square$