We note this system of equations is equivalent to evaluating the polynomial (in $a$ ) $P(a)=w a+x a^{2}+y a^{3}+z a^{4}$ at $1,2,3$, and 4 . We know that $P(0)=0, P(1)=5, P(2)=7, P(3)=11$, $P(4)=1$. The finite difference of a polynomial $f$ is $f(n+1)-f(n)$, which is a polynomial with degree one less than the degree of $f$. The second, third, etc finite differences come from applying this operation repeatedly. The fourth finite difference of this polynomial is constant because this is a fourth degree polynomial. Repeatedly applying finite differences, we get $$\begin{array}{lccccccc} 0 & & 5 & & 7 & & 11 & \\ & 5 & 2 & & 4 & & -10 \\ & -3 & & 2 & & -14 & \\ & & 5 & & -16 & & \\ & & & & -21 & & \end{array}$$ and we see that the fourth finite difference is -21 . We can extend this table, knowing that the fourth finite difference is always -21 , and we find that that $P(5)=-60$. The complete table is $$\begin{array}{ccccccccccc} 0 & & 5 & & 7 & & 11 & & 1 & & -60 \\ & 5 & & 2 & & 4 & & -10 & & -61 & \\ & -3 & & 2 & & -14 & & -51 & & \\ & & 5 & & -16 & & -37 & & & \\ & & & -21 & & -21 & & & & \end{array}$$