Eliminate

Eliminate[eqns,vars]

eliminates variables between a set of simultaneous equations.

Details and Options

  • Equations are given in the form lhs==rhs.
  • 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.

Examples

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Basic Examples  (1)

Eliminate the variable between two equations:

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)

InverseFunctions  (3)

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:

Mode  (1)

Find a modulus for which a system of equations has a solution and eliminate a variable:

WorkingPrecision  (1)

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:

Properties & Relations  (3)

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:

Possible Issues  (1)

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:

Introduced in 1988
 (1.0)