# 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

open allclose all

## Basic Examples(2)

Eliminate the variable between two equations:

Eliminate multiple variables in a system of equations:

## Scope(7)

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:

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

## Options(5)

### InverseFunctions(3)

By default, Eliminate uses inverse functions but prints warning messages:  With , Eliminate does not print inverse function messages:

With , 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: