An algebra problem by mridul jain

Algebra Level 4

Find $a_{}^{}x^5 + b_{}y^5$ if the real numbers $a_{}^{}$ , $b_{}^{}$ , $x_{}^{}$ , and $y_{}^{}$ satisfy the equations $\begin{aligned} ax + by &= 3^{}_{}\\ ax^2 + by^2 &= 7^{}_{}\\ ax^3 + by^3 &= 16^{}_{}\\ ax^4 + by^4 &= 42^{}_{}. \end{aligned}$