Solution:

Answer: $-60$

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

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 $P(5) = -60$.

The complete table is