# Solve the system of nonlinear equations

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 $y-5z=1$.

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 $x=22$.
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.
The text passage contains a mathematical question in the form of a system of nonlinear equations to be solved for real number \( x \) and prime numbers \( y \) and \( z \). The passage also provides a detailed solution to the problem, including the steps taken and the final values of \( x \), \( y \), and \( z \).
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Solve the system of nonlinear equations

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 $y-5z=1$.

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 $x=22$.
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.
user
The text passage contains a mathematical question in the form of a system of nonlinear equations to be solved for real number \( x \) and prime numbers \( y \) and \( z \). The passage also provides a detailed solution to the problem, including the steps taken and the final values of \( x \), \( y \), and \( z \).

TYPE: SINGLE_QUESTION_WITH_ANSWER