Suppose we have two random variables $X$ and $Y$ with cumulative distribution functions $F(x)$ and $G(y)$. We know that $Y = aX + b$.

I want to compute $Z(x) = F(x) - G(y)$.

What I have so far is the following: $$F(x) = \int_{-\infty}^{x}f(x)dx $$

$$G(y) = \int_{-\infty}^{y}g(y)dy = \int_{-\infty}^{y}\frac{1}{|a|}f\left(\frac{y-b}{a}\right)dy $$

$$Z(x) = \int_{-\infty}^{x}f(x)dx - \int_{-\infty}^{y}\frac{1}{|a|}f(x)dy $$

How should I go on doing the subtraction of the integrals? Is this even possible as the integrales belong to different variables (dx and dy)?