An algebra problem by mridul jain

Algebra Level 5

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