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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 xy-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]= 
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