text: If the perimeter of a rectangle is 5x^2+2xy-7y^2+16, and the length is -2x^2+6xy-y^2+15, what is the width?
  
  Oct 27, 2016
  
  $\Rightarrow W = \frac{9}{2} {x}^{2} - 5 x y - \frac{5}{2} {y}^{2} - 7$
  
  Explanation:
  
  Name $P$ the perimeter of the rectangle.
  Name $L$ its length and $W$ its width.
  
  Given:
  $P = 5 {x}^{2} + 2 x y - 7 {y}^{2} + 16$
  
  $L = - 2 {x}^{2} + 6 x y - {y}^{2} + 15$
  
  ${P}_{R e c} = 2 \cdot \left(L + W\right)$
  
  $\Rightarrow 5 {x}^{2} + 2 x y - 7 {y}^{2} + 16 = 2 \cdot \left(- 2 {x}^{2} + 6 x y - {y}^{2} + 15 + W\right)$
  
  $\Rightarrow 5 {x}^{2} + 2 x y - 7 {y}^{2} + 16 = - 4 {x}^{2} + 12 x y - 2 {y}^{2} + 30 + 2 W$
  
  $\Rightarrow 5 {x}^{2} + 2 x y - 7 {y}^{2} + 16 + 4 {x}^{2} - 12 x y + 2 {y}^{2} - 30 = + 2 W$
  
  $\Rightarrow 5 {x}^{2} + 4 {x}^{2} + 2 x y - 12 x y - 7 {y}^{2} + 2 {y}^{2} + 16 - 30 = + 2 W$
  
  $\Rightarrow 9 {x}^{2} - 10 x y - 5 {y}^{2} - 14 = + 2 W$
  
  $\Rightarrow \frac{9 {x}^{2} - 10 x y - 5 {y}^{2} - 14}{2} = W$
  
  $\Rightarrow \frac{9}{2} {x}^{2} - 5 x y - \frac{5}{2} {y}^{2} - 7 = W$
explanation: The document contains a math question and its corresponding answer. The question asks for the width of a rectangle given its perimeter and length, and the answer provides a detailed step-by-step solution to find the width.
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: If the perimeter of a rectangle is 5x^2+2xy-7y^2+16, and the length is -2x^2+6xy-y^2+15, what is the width?
  
  Oct 27, 2016
  
  $\Rightarrow W = \frac{9}{2} {x}^{2} - 5 x y - \frac{5}{2} {y}^{2} - 7$
  
  Explanation:
  
  Name $P$ the perimeter of the rectangle.
  Name $L$ its length and $W$ its width.
  
  Given:
  $P = 5 {x}^{2} + 2 x y - 7 {y}^{2} + 16$
  
  $L = - 2 {x}^{2} + 6 x y - {y}^{2} + 15$
  
  ${P}_{R e c} = 2 \cdot \left(L + W\right)$
  
  $\Rightarrow 5 {x}^{2} + 2 x y - 7 {y}^{2} + 16 = 2 \cdot \left(- 2 {x}^{2} + 6 x y - {y}^{2} + 15 + W\right)$
  
  $\Rightarrow 5 {x}^{2} + 2 x y - 7 {y}^{2} + 16 = - 4 {x}^{2} + 12 x y - 2 {y}^{2} + 30 + 2 W$
  
  $\Rightarrow 5 {x}^{2} + 2 x y - 7 {y}^{2} + 16 + 4 {x}^{2} - 12 x y + 2 {y}^{2} - 30 = + 2 W$
  
  $\Rightarrow 5 {x}^{2} + 4 {x}^{2} + 2 x y - 12 x y - 7 {y}^{2} + 2 {y}^{2} + 16 - 30 = + 2 W$
  
  $\Rightarrow 9 {x}^{2} - 10 x y - 5 {y}^{2} - 14 = + 2 W$
  
  $\Rightarrow \frac{9 {x}^{2} - 10 x y - 5 {y}^{2} - 14}{2} = W$
  
  $\Rightarrow \frac{9}{2} {x}^{2} - 5 x y - \frac{5}{2} {y}^{2} - 7 = W$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding answer. The question asks for the width of a rectangle given its perimeter and length, and the answer provides a detailed step-by-step solution to find the width.
  
  Label: question-and-answer