sum of linear equation system solution

Question

if I sum 2 solutions of a non-homogeneous linear equation system would it still be an answer, and if I subtract 2 solutions of a homogeneous linear equation would it still be an answer? thanks in advance.

Answer 1

Consider the non-homogeneous linear equation:

$x + y = 5$

Both $(5, 0) $ and $(0, 5) $ are solutions. But $(5, 0) + (0, 5) = (5, 5) $ which is not a solution.

On the other hand, let $T(x_1, \cdots, x_n) $ be a homogeneous linear equation on $n $ variables and let $s_1, s_2$ be two solutions.

Then $T(s_1 - s_2) = T(s_1) + T(-s_2) = T(s_1) - T(s_2) = 0 - 0 = 0$ giving that $s_1 - s_2$ is also a solution.

What you can do however is sum a solution of the non-homogeneous to a solution of the homogeneous and still get a solution for the non-homogeneous:

Let $T(x_1, \cdots, x_n) $ represent a linear system. Setting $T(x_1, \cdots, x_n) = 0$ represents the homogeneous system. Let $h$ denote a solution of the homogeneous system i.e. $T(h) = 0$. Setting $T(x_1, \cdots, x_n) = k, k\not=0$ represents a non-homogeneous system. Let $s$ be a solution of $T(x_1, \cdots, x_n) = k$. Thus $T(s) = k $. Now we see that $s + h $ is a solution of the same non-homogeneous system:

$T(s + h) = T(s) + T(h) = k + 0 = k $

A similar construct yields that if $s_1, s_2$ are solutions of the non-homogeneous system, then $s_1 - s_2$ is a solution of the homogeneous system:

$T(s_1 - s_2) = T(s_1) + T(-s_2) = T(s_1) - T(s_2) = k - k = 0$

Hence the theorem: "All solutions of a non-homogeneous linear system are given by adding up one particular solution to the general solution of the homogeneous system". That is proven by the two calculations above! 1) adding a homogeneous solution to a non-homogeneous yields a non-homogeneous. 2) subtracting two non-homogeneous solutions yields a homogeneous solution.