Hi I am trying to complete the following question in my practice:
Suppose a, b, c are integers and x, y and z are non-zero real numbers that satisfy the following equations:
\begin{equation} \cfrac{xy}{x+y} = a \quad and \quad \cfrac{xz}{x+z} = b \quad and \quad \cfrac{yz}{y+z} = c \end{equation}
Prove that $ x $ is rational
Source: Discrete Mathematics with Applications 4th edition
I tried combining the equations and logically conclude that $ x $ is rational but to no avail as I do not know how to separate $ x $ from the rest of the variables. The only starting point that I have is the theorem for rational numbers:
\begin{equation} if \; x \in \mathbb{Q}, \; x = \cfrac{a}{b} \; where \; a,b \in \mathbb{Z} \; and \; b \neq 0 \end{equation}
I have no other idea how to carry on from here. Could someone please advise me?