This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Eliminate

 Eliminateeliminates variables between a set of simultaneous equations.
• Equations are given in the form .
• Simultaneous equations can be combined either in a list or with .
• A single variable or a list of variables can be specified.
• Variables can be any expressions.
• Eliminate works primarily with linear and polynomial equations.
Eliminate the variable between two equations:
Eliminate the variable between two equations:
 Out[1]=
 Scope   (6)
A system of linear equations:
A system of polynomial equations:
Eliminate two variables:
A system of equations involving radicals:
A system of transcendental equations:
A system of modular equations:
 Options   (5)
By default, Eliminate uses inverse functions but prints warning messages:
With InverseFunctions->True, Eliminate does not print inverse function messages:
With InverseFunctions->False, Eliminate does not use inverse functions:
Eliminating variables from algebraic equations does not require using inverse functions:
Find a modulus for which a system of equations has a solution and eliminate a variable:
By default, Eliminate computes with exact coefficients:
This performs the elimination using 20-digit approximate number coefficients:
 Applications   (2)
Rewrite in terms of and :
Find a condition for two polynomials to have a common root:
This solves the same problem using Resolve:
The condition is equivalent to Resultant of the polynomials being zero:
Equations returned by Eliminate do not contain the elimination variables:
Equations returned by Eliminate are implied by the input equations:
Use Resolve to check this property:
Use Resolve to eliminate an existential quantifier:
Eliminate gives the same set of equations, but does not give inequations:
Eliminate a variable using GroebnerBasis:
Eliminate a variable from a pair of polynomials using Resultant:
Use Reduce to eliminate an existential quantifier and solve the resulting system:
When the input contains only equations, Eliminate returns only equations:
The zero set of the result is the Zariski closure of the projection of the zero set of eqns:
Use Resolve to get a result with zero set equal to the projection of the zero set of eqns:
When the input contains inequations, Eliminate returns equations and inequations:
The zero set of the result is equal to the projection of the zero set of eqin:
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