Can i substitute same equation into another in two different forms?

Question

Suppose I have the following two equalities: $$ \frac{x}{c}=\frac{y}{d} \quad \text{ and } x-a*c = y-a*d $$ with $a\in (0,1)$.

Suppose I divide both sides of the second equality by $c$, and get $$ \frac{x}{c}-a = \frac{y}{c} - a*\frac{d}{c} $$

Can I substitute $\frac{x}{c}=\frac{y}{d}$ into the LHS and $\frac{d}{c}=\frac{y}{x}$ into the RHS

(ignore why I would do some substitution. Im just wondering if it is valid)

Answer 1

Yes, you may use your equation to take advantage of $$\frac{x}{c}=\frac{y}{d}$$ and $$\frac{d}{c}=\frac{y}{x}$$

You have both equations and you may use both of them as you wish.

Answer 2

Both your substitutions are valid since these two equations are simultaneous.