Math Question - Find unknown variable with equations

Question

Three-digit numbers P, Q, and R, QQP and PQQ are formed using the single digits P, Q and R such that:

PQR + 2 = PQQ

and

PQR * 2 = QQP

Determine all possible combinations of values of P, Q and R that satisfy the two equations.

Here is my work so far:

What we know:

R+2=Q, PQ remains the same meaning there were no carrying needed. R+2<=9

The second equation, R*2=P. R cannot be 0 as anything multiplied by 0 is 0.

What we know about R:

Ranges from 1 – 7

What we know about P:

Cannot be 0 as it is in the hundred digits. 
P*2 = Q

What we know about Q:

Q*2 = the first digit of QR.

Answer 1

Hint: The first equation gives us $ R + 2 = Q$.

Hint: The second equation then gives us $199 P - 4 = 88Q$

Do some work here, yes you.

Hence conclude that $P = 4, Q = 9, R = 7$.