Can the Laplace Inverse be Applied to Divided Transfer Functions?

[Chacabucogod]

I was wondering whether this can be done:

Let's say you have transfer function that goes like this:

$$\frac{Y(s)}{U(s)}= \frac{N(s)}{D(s)}$$

Now let's say I divide my transfer into two:

$$\frac{Y(s)}{Z(s)}= N(s)$$

$$\frac{Z(s)}{U(s)}= \frac{1}{D(s)}$$

Can I apply the Laplace Inverse to these two equation separately and then substitute the value of z(t) on one?

$$D(s)Z(s)=U(s)$$

$$N(s)Z(s)=Y(s)$$

Thank you!

 

[Hesch]

Of course you can:

You then have:

Y(s)/U(s) = (N(s)*Z(s) ) / (D(s)*Z(s) ) => (shorten right fraction)

Y(s)/U(s) = N(s)/D(s)

No problem, assuming that Z(s) ≠ 0.