Solve the system of nonlinear equations

Question

Solve the following system of equations for real number $x$ and the prime numbers $y$ and $z$.

$$ \left\{ \begin{array}{c} \frac{1}{x}+\frac{5}{y}-\frac{1}{z}=0 \\ x-yz=0 \\ \end{array} \right. $$

I started to do this problem by finding the common denominator but I stuck. I think that I have to change this system to a simple linear system of equations, but I don't know how to do it. In the last step, I found an equation which is $y-5z=1$.

Answer 1

The equation you found should be $y-5z=1$, hence $y = 1 + 5z$. Note that $y, z$ are prime numbers, if $z\neq 2$ then $y$ must be even, which means $y=2$, and this is impossible. Therefore, $z=2$ , $y=11$ and $x=22$.

Answer 2

The equation that you found is a little bit wrong: after bringing the first equation over a common denominator, you should get that $y-5z=1$, not zero. This can be rewritten as $y=1+5z$. Now, remembering that $y$ and $z$ are prime numbers, consider two cases: $z=2$ or $z$ is an odd prime.