# Eliminating Variables

When you write down a set of simultaneous equations in *Mathematica*, you are specifying a collection of constraints between variables. When you use Solve, you are finding values for some of the variables in terms of others, subject to the constraints represented by the equations.

Solve[eqns,vars,elims] | find solutions for vars, eliminating the variables elims |

Eliminate[eqns,elims] | rearrange equations to eliminate the variables elims |

Eliminating variables.

Here are two equations involving

,

and the "parameters"

and

.

Out[1]= | |

If you solve for both

and

, you get results in terms of

and

.

Out[2]= | |

Similarly, if you solve for

and

, you get results in terms of

and

.

Out[3]= | |

If you only want to solve for

, however, you have to specify whether you want to eliminate

or

or

. This eliminates

, and so gives the result in terms of

and

.

Out[4]= | |

If you eliminate

, then you get a result in terms of

and

.

Out[5]= | |

In some cases, you may want to construct explicitly equations in which variables have been eliminated. You can do this using Eliminate.

This combines the two equations in the list

, by eliminating the variable

.

Out[6]= | |

This is what you get if you eliminate

instead of

.

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As a more sophisticated example of Eliminate, consider the problem of writing in terms of the "symmetric polynomials" and .

To solve the problem, we simply have to write

in terms of

and

, eliminating the original variables

and

.

Out[8]= | |

In dealing with sets of equations, it is common to consider some of the objects that appear as true "variables", and others as "parameters". In some cases, you may need to know for what values of parameters a particular relation between the variables is *always* satisfied.

SolveAlways[eqns,vars] | solve for the values of parameters for which the eqns are satisfied for all values of the vars |

Solving for parameters that make relations always true.

This finds the values of parameters that make the equation hold for all

.

Out[9]= | |

Out[10]= | |

This finds values of the undetermined coefficients.

Out[11]= | |