**GNU Emacs Calc 2.02 Manual**

### 11.6.2 Solving Systems of Equations

You can also use the commands described above to solve systems of
simultaneous equations. Just create a vector of equations, then
specify a vector of variables for which to solve. (You can omit
the surrounding brackets when entering the vector of variables
at the prompt.)

For example, putting ``[x + y = a, x - y = b]'` on the stack
and typing `a S x,y ``RET` produces the vector of solutions
``[x = a - (a-b)/2, y = (a-b)/2]'`. The result vector will
have the same length as the variables vector, and the variables
will be listed in the same order there. Note that the solutions
are not always simplified as far as possible; the solution for
x here could be improved by an application of the `a n`
command.

Calc's algorithm works by trying to eliminate one variable at a
time by solving one of the equations for that variable and then
substituting into the other equations. Calc will try all the
possibilities, but you can speed things up by noting that Calc
first tries to eliminate the first variable with the first
equation, then the second variable with the second equation,
and so on. It also helps to put the simpler (e.g., more linear)
equations toward the front of the list. Calc's algorithm will
solve any system of linear equations, and also many kinds of
nonlinear systems.

Normally there will be as many variables as equations. If you
give fewer variables than equations (an "over-determined" system
of equations), Calc will find a partial solution. For example,
typing `a S y ``RET` with the above system of equations
would produce ``[y = a - x]'`. There are now several ways to
express this solution in terms of the original variables; Calc uses
the first one that it finds. You can control the choice by adding
variable specifiers of the form ``elim(``v`)' to the
variables list. This says that `v` should be eliminated from
the equations; the variable will not appear at all in the solution.
For example, typing `a S y,elim(x)` would yield
``[y = a - (b+a)/2]'`.

If the variables list contains only `elim`

specifiers,
Calc simply eliminates those variables from the equations
and then returns the resulting set of equations. For example,
`a S elim(x)` produces ``[a - 2 y = b]'`. Every variable
eliminated will reduce the number of equations in the system
by one.

Again, `a S` gives you one solution to the system of
equations. If there are several solutions, you can use `H a S`
to get a general family of solutions, or, if there is a finite
number of solutions, you can use `a P` to get a list. (In
the latter case, the result will take the form of a matrix where
the rows are different solutions and the columns correspond to the
variables you requested.)

Another way to deal with certain kinds of overdetermined systems of
equations is the `a F` command, which does least-squares fitting
to satisfy the equations. See section 11.8 Curve Fitting.