text: This is documentation for Mathematica 4, which was
  based on an earlier version of the Wolfram Language.
  Wolfram Research, Inc.
  
  3.4.9 Solving Equations with Subsidiary Conditions
  
  In doing calculations with equations, you will often find it convenient to think of a particular set of equations as the "main" ones you are working with, and to think of other equations as "subsidiary conditions" that must also be satisfied.
  
  A typical thing to do is to assign a name to the list of subsidiary conditions, and then to include them by name in each list of equations you give to Solve.
  
  sincos is defined to be the equation .
  
  In[1]:= sincos = Sin[x]^2 + Cos[x]^2 == 1
  
  Out[1]=
  
  This solves the equation , with the "subsidiary condition" .
  
  In[2]:= Solve[ { Sin[x] + 2 Cos[x] == 1, sincos } ,
  
  { Sin[x], Cos[x] } ]
  
  Out[2]=
  
  Here is another equation, solved with the same "subsidiary condition".
  
  In[3]:= Solve[ { Sin[x] == Cos[x], sincos } ,
  
  { Sin[x], Cos[x] } ]
  
  Out[3]=
  
  When you are working with polynomials, there are ways to use GroebnerBasis and PolynomialReduce to reduce sets of polynomials subject to subsidiary conditions.
  
  This sets up a Gröbner basis.
  
  In[4]:= g = GroebnerBasis[{x + y - a, x y - b}, {x, y}]
  
  Out[4]=
  
  This gives a reduction of x^3+y^3 in terms of the elements of g.
  
  In[5]:= PolynomialReduce[x^3 + y^3, g, {x, y}]
  
  Out[5]=
  
  This shows how x^3+y^3 can be reduced if x+y-a and x y-b are assumed to be zero.
  
  In[6]:= Last[%]
  
  Out[6]=
  
  This reduces x^4+y^4 with the same subsidiary condition.
  
  In[7]:= Last[PolynomialReduce[x^4 + y^4, g, {x, y}]]
  
  Out[7]=
explanation: The document is a section from Mathematica 4 documentation, focusing on solving equations with subsidiary conditions. It provides examples of how to define and use subsidiary conditions in equations, as well as how to use GroebnerBasis and PolynomialReduce for polynomial reductions under these conditions. The text contains multiple examples and their corresponding outputs, but it does not contain a single, isolated question or a question-and-answer pair. Instead, it is a series of instructional examples and explanations.
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: This is documentation for Mathematica 4, which was
  based on an earlier version of the Wolfram Language.
  Wolfram Research, Inc.
  
  3.4.9 Solving Equations with Subsidiary Conditions
  
  In doing calculations with equations, you will often find it convenient to think of a particular set of equations as the "main" ones you are working with, and to think of other equations as "subsidiary conditions" that must also be satisfied.
  
  A typical thing to do is to assign a name to the list of subsidiary conditions, and then to include them by name in each list of equations you give to Solve.
  
  sincos is defined to be the equation .
  
  In[1]:= sincos = Sin[x]^2 + Cos[x]^2 == 1
  
  Out[1]=
  
  This solves the equation , with the "subsidiary condition" .
  
  In[2]:= Solve[ { Sin[x] + 2 Cos[x] == 1, sincos } ,
  
  { Sin[x], Cos[x] } ]
  
  Out[2]=
  
  Here is another equation, solved with the same "subsidiary condition".
  
  In[3]:= Solve[ { Sin[x] == Cos[x], sincos } ,
  
  { Sin[x], Cos[x] } ]
  
  Out[3]=
  
  When you are working with polynomials, there are ways to use GroebnerBasis and PolynomialReduce to reduce sets of polynomials subject to subsidiary conditions.
  
  This sets up a Gröbner basis.
  
  In[4]:= g = GroebnerBasis[{x + y - a, x y - b}, {x, y}]
  
  Out[4]=
  
  This gives a reduction of x^3+y^3 in terms of the elements of g.
  
  In[5]:= PolynomialReduce[x^3 + y^3, g, {x, y}]
  
  Out[5]=
  
  This shows how x^3+y^3 can be reduced if x+y-a and x y-b are assumed to be zero.
  
  In[6]:= Last[%]
  
  Out[6]=
  
  This reduces x^4+y^4 with the same subsidiary condition.
  
  In[7]:= Last[PolynomialReduce[x^4 + y^4, g, {x, y}]]
  
  Out[7]=
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a section from Mathematica 4 documentation, focusing on solving equations with subsidiary conditions. It provides examples of how to define and use subsidiary conditions in equations, as well as how to use GroebnerBasis and PolynomialReduce for polynomial reductions under these conditions. The text contains multiple examples and their corresponding outputs, but it does not contain a single, isolated question or a question-and-answer pair. Instead, it is a series of instructional examples and explanations.
  
  Label: multiple-question-and-answer