Specified a system of equations, (1), we will have one of the three probabilities for the number of solutions.

1.   No solution.

2.   Accurately one solution.

3.   Infinitely many solutions.

Before going on to the other section we require to see at one more situations. The system of equations in (1) is termed as a non-homogeneous system whether at least one of the bi's is not zero. If though all of the bi's are zero we identify the system as homogeneous and the system will be as,

a₁₁ x₁ + a₁₂ x₂ +................+a_1n x_n = 0

a₂₁ x₁ + a₂₂ x₂ +.............. +a_2n x_n  = 0

...................

a_n1 x₁ + a_n2 x₂ +............... +a_nn x_n  = 0                                 ...................(2)

Now, notice that in the homogeneous case we are guaranteed to have the following solution.

x₁ + x₂+....... +x_n  = 0

This solution is frequently termed as the trivial solution.

The fact given above can be modified to the following, for homogeneous systems.