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
.
| Out[7]= |  |
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]= |  |
