text: # 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.
explanation: The document contains a math question and a detailed solution to the problem. The question asks to solve a system of nonlinear equations for real number \( x \) and prime numbers \( y \) and \( z \). The solution provided includes the steps taken to solve the system, leading to the final values \( x = 22 \), \( y = 11 \), and \( z = 2 \).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed solution to the problem. The question asks to solve a system of nonlinear equations for real number \( x \) and prime numbers \( y \) and \( z \). The solution provided includes the steps taken to solve the system, leading to the final values \( x = 22 \), \( y = 11 \), and \( z = 2 \).
  
  Label: question-and-answer