I became aware of this question by way of an answer on Meta and feel I must push back against the comment of DonAntonio and the answer of amWhy.
Uniqueness of the solution of the system
$$
\begin{aligned}
x_1&=3\\
x_2&=2\\
x_3&=3
\end{aligned}
$$
is obvious and needs no proof. What is there to prove? Is it conceivable that if you plug in numbers other than $3,$ $2,$ and $3$ for $x_1,$ $x_2,$ and $x_3$ you might obtain three true statements?
The determinant is a complicated object, and by bringing it in in this situation, you are making something simple appear much more difficult than it actually is.
Here's what I think you were probably getting at: Let's start with a simpler analogue. Is the solution of the equation $x=2$ unique? Of course is is: $2$ is the only solution. Now $x=2$ may be the end result of simplifying a more complicated equation, such as $13x=26.$ The latter is a special case of the general equation $ax=b.$ It is certainly the case that the latter has a unique solution if and only if $a\ne0.$ If $a=0,$ then there is no solution unless $b=0,$ in which case there are infinitely many solutions.
Likewise, the matrix equation $Ax=b,$ where $A$ is a square matrix and $x$ and $b$ are column vectors, has a unique solution if and only if $\det A\ne0.$ If $\det A=0,$ then it has either no solution or infinitely many solutions.
So it is helpful to introduce the determinant to make statements about the nature of the solution set of the general equation $Ax=b.$ But for concrete $A$ and $b,$ it is usually more efficient to row reduce the system than to compute $\det A.$ (More precisely, computing $\det A$ is best done by actually performing row reduction, but there is no need to mention determinants if you are row reducing to solve a concrete problem.) The end result of the row-reduction process will tell you whether there is a unique solution or not.
The only thing that might need proof is that the three row operations (swapping rows, multiplying a row by a non-zero number, adding a multiple of one row to another row) preserve the solution set. That is generally proved in a linear algebra course, and you can probably assume it from that point on. If not, let $S$ be a system and let $S'$ be the system that results from applying a row operation. You just need to prove that any solution to $S$ is a solution to $S',$ and that any solution to $S',$ is a solution to $S.$ This is straightforward, but it seems like overkill to do it in every row reduction problem you perform.
If you are asking for a good way to prove that the solution set isn't a vector space, I think the best way is to notice that it doesn't contain the zero vector.
EDIT: On re-reading the question, I think you are saying that you know the solution set isn't a vector space, you want to prove that if $x_1$ and $x_2$ are solutions then $x_1+x_2$, $ax_1$, and $a_1x_1+a_2x_2$ aren't solutions. Perhaps the easiest way to handle this is to look at the non-homogeneous system as a matrix equation $Ax=b$ with $b$ not the zero vector. Then we are assuming $$Ax_1=b,\qquad Ax_2=b$$ Thus, we have $$A(x_1+x_2)=Ax_1+Ax_2=b+b\ne b$$ since we are assuming $b\ne0$, so this proves $x_1+x_2$ is not a solution. We have $$A(ax_1)=aAx_1=ab\ne b$$ again since $b\ne0$ (of course, we have to assume here $a\ne1$). Finally, $$A(a_1x_1+a_2x_2)=a_1Ax_1+a_2Ax_2=a_1b+a_2b=(a_1+a_2)b$$ So $a_1x_1+a_2x_2$ can be a solution; indeed, it will be a solution, if and only if $a_1+a_2=1$. There's a name for this: if $a_1+a_2=1$, then $a_1x_1+a_2x_2$ is called a convex linear combination of $x_1$ and $x_2$, and the solution set of a non-homogeneous system is closed under taking convex linear combinations.
Best Answer
Let $\bar{x}$ be a solution to $Ax=b$ and $v$ be a solution to $Ax=0$. From here, it follows
$$A\bar{x} + Av = b$$
$$A(\bar{x}+v) = b$$
So, $\bar{x}+v$ is also a solution to $Ax=b$. In other words a solution to non-homogeneous system plus the solution to homogeneous system is also a solution to non-homogeneous system.
There is no relation between $x_1$ and $x_2$ since they are solutions to two different systems which happen to have the same $A$. In geometry point of view, think of $A$ as a transformation which transforms two arbitrary vectors $x_1$ and $x_2$ to two different vectors $b_1$ and $b_2$.