Using Gauss-Jordan method to indicate the solutions:

Consider the given system of linear equations,

x1+x2=2x2+x3=3x1+2x2+x3=5

The augmented matrix of this system is as follows:

A|b=[1 1 00 1 11 2 1][235]

The Gauss-Jordan method is applied to find the solutions of the above system of linear equations.

Replace row 3 of A|b by (row 3 – row 1), then the following matrix is obtained,

A1|b1=[1 1 00 1 10 1 1][233]

Now, replace row 3 of A1|b1 by (row 3 - row 2), then the following matrix is obtained,

A2|b2=[1 1 00 1 10 0 0][230]

This produces the following result,

x1+x2=2x2+x3=3

Solving the above equations, the following result is obtained,

x1=k−1x2=3−kx3=k

Therefore, the above system of linear equations has a unique solution.