A linear system of equations is a set of linear equations in
variables (sometimes called "unknowns"). Linear
systems can be represented in matrix form as the matrix
equation

If ,
then the system is (in general) overdetermined and there is no solution.

If
and the matrix
is nonsingular, then the system has a unique solution in the variables. In particular, as shown by Cramer's
rule, there is a unique solution if has a matrix inverse .
In this case,

(2)

If ,
then the solution is simply . If has no matrix inverse, then
the solution set is the translate of a subspace of dimension
less than
or the empty set.

If two equations are multiples of each other, solutions are of
the form

(3)

for
a real number. More generally, if , then the system is underdetermined. In this case, elementary row and column operations
can be used to solve the system as far as possible, then the first components can be solved in terms of the last components to find the solution space.