AIME 1990 Problem 15

Find $ax^5 + by^5$ if the real numbers $a,b,x,$ and $y$ satisfy the equations \begin{align*} ax + by &= 3, \ ax^2 + by^2 &= 7, \ ax^3 + by^3 &= 16, \ ax^4 + by^4 &= 42. \end{align*}