Probability of x modulus a prime = a

Question

I am working on a project that involves analyzing the distribution of simple hash functions. One such construction relies on the fact that given a primes $p,q$, the unit group for $p$ $U_p$, a random integer $x$ and random unit $a\in U_p$, I want to show that the probability distribution of $x \cdot q \equiv a~mod~p$ is uniform, with parameter $\frac{1}{p}$ I am fairly certain in relies on the randomness of $x$ but am having trouble formally proving this.