## Difference Between Multiple Solutions of Non Homogeneous Linear Equations

Suppose we have a system of
$n$
non homogeneous linear equations
$\sum^m_{j=1} a_{ij}x_j=b_i , \: i= 1,...,n$
.
Suppose we have two solutions
$(x_1,...,x_m)$
and
$(x'_1,...,x'_m)$
.
The difference between these two solutions,
$(x'_1-x_1,...,x'_m-x_m)$
is a solution to the homogeneous system of equations
$\sum^m_{j=1} a_{ij}x_j=0 , \: i= 1,...,n$
.
Notice that these solutions satisfy
$\sum^m_{j=1} a_{ij}x_j=b_i , \: \sum^m_{j=1} a_{ij}x'_j=b_i, \: i= 1,...,n$
.
Subtracting these gives the homogeneous system
$\sum^m_{j=1} a_{ij}(x'_j-x_j)=0, \: i= 1,...,n$
which is equivalent to the system
$\sum^m_{j=1} a_{ij}x_j=0 , \: i= 1,...,n$
.