Find impulse response when given output and another input

Hello, I was given the following objective question to be solved with MATLAB:

Suppose that an input signal is given as ?[?] = cos(0.1??) for 0 ≤ ? ≤ 20. The signal ?[?] was then distorted by a noise and changed into the signal x?[?] = cos(0.1??) + cos(0.5??) for 0 ≤ ? ≤ 20. Given a difference equation of a filter, ?[?] = 0.5?[?] + 0.5?[? − ?], find its impulse response, ℎ[?] and perform the following tasks (use ? = 2).

The lecture notes say to use convolution to solve, but I don't see how if we are given the output and 1/2 inputs. My understanding of transfer functions is that I should convert the given xs[n] and y[n] to the z-doman and then divide Y(z)/X(z) to obtain H(z). The convert back to h[n]. I have tried a number of ways to solve it this way for h[n], but I'm stuck and my results don't seem to be correct. I cannot figure out how to make ztrans or iztrans compatible with a double vector, not a syms vector. Please give some direction on how to approach this problem.