MHB Solve Algebra Question Easily: 34

[Ilikebugs]

View attachment 6159I know that I can use guess and check, but I was wondering if there was an easier way? I got 34

 

[I like Serena]

I know that I can use guess and check, but I was wondering if there was an easier way? I got 34

Hey Ilikebugs! Nice problem! ;)

I'll put my solution in spoiler tags for other people who may like the problem as well.

Spoiler

First the observation: $P\ne 0$ and $Q \ne 0$, because otherwise we divide by zero.
Note that we can always multiply or divide both sides by a non-zero value, but if the value can be zero, we have to check.

Then it follows that:

[math]
\frac PQ - \frac QP = \frac{P+Q}{PQ}
\quad\Rightarrow\quad \frac{P^2-Q^2}{PQ} = \frac{P+Q}{PQ}
\quad\Rightarrow\quad (P+Q)(P-Q)=P+Q \\
\quad\Rightarrow\quad P+Q=0 \quad\textit{ or }\quad P-Q=1
\quad\Rightarrow\quad Q=-P \quad\textit{ or }\quad Q=P-1
[/math]

Considering that neither P nor Q can be zero, the first condition gives us 18 solutions, and the second 16 solutions, for a total of 34.