Or, if you don't have a

flashy calculator...

Given a system of

linear equations:

a

_{1,1}x

_{1} + a

_{1,2}x

_{2}
... + a

_{1,n} = b

_{1}
a

_{2,1} + ... = b

_{2}
.

.

.

a

_{m,1} + ... = b

_{m}
We can represent this system as a

matrix equation:

**A . X ** =

**B**
Where A is the m*n

coefficient matrix, whose (i,j) entry is
a

_{i,j},

X is [ x

_{1}, x

_{2}, ... , x

_{n} ] ,

and B is [ b

_{1}, b

_{2}, ... , b

_{m} ]

Then, we define the

** augmented matrix ** of this system to be [

**A|B** ]

(i.e. an m*(n+1) matrix given by adding

**B** as an extra column to the right side of

**A**)

This matrix can now be transformed, using

elementary row operations, into

row reduced echelon form.

Given a

row-reduced matrix

**E**, we can then use
the

Gauss-Jordan procedure to find solutions.

**Case 1: ** The last non-zero row of

**E** is 0,0,...,1. In this case, the system is inconsistent.

**Case 2:** **E** has n non-zero rows. Then (x

_{1} , x

_{2}, ... , x

_{n}) = the rightmost column of

**E**.

**Case 3:** **E** has k non-zero rows. Assign

parametric values to the n-k variables whose rows in

**E** are zero.