Lets say I have the following set of equations:
$$\begin{matrix}-x_1&+&2x_2&+&5x_3&-&2x_4&=&0\\ -2x_1&+&4x_2&+&10x_3&-&2x_4&=&4\\ -x_1&+&2x_2&+&2x_3&-&2x_4&=&-3\\ 2x_1&-&4x_2&-&7x_3&+&5x_4&=&5\end{matrix}$$
the solution would be:
$$L = \{ (1 + 2λ, λ, 1, 2) ~|~ λ \in\mathbb R\}$$
this can be decomposed into:
$$L = \{(1, 0, 1, 2) + λ (2,1,0,0)~\vert~\lambda\in\mathbb R\}$$
whereas the first vector is an independent solution vector for $λ = 0$ and the second vector is representing the homogenous solution.
But why exactly is $λ (2,1,0,0)$ representing a homogenous solution? What exactly does this part of the solution represent?
As a consequence of Rank–nullity theorem, for a linear system $Ax=b$ with $m$ equations in $n$ unknowns the solutions for the homogeneous system $Ax_H=0$ belong to a subspace with dimension $n-r$ with $r=\operatorname{rank} (A)$.
When a particular solution $x_P$ for the system exists, that is when $b\in \operatorname{Col}(A)$), such that $Ax_P=b$ we also have that
$$A(x_P+x_H)=Ax_P+Ax_H=b+0=b$$
therefore all the solutions for the system can be expressed in the form
$$x=x_P+x_H$$
beeing $x_P$ one particular solution for the system $Ax=b$ and $x_H$ one solution for the homogeneous system $Ax=0$.
Since $x_H$ belong to a subspace, it is usually represented as a linear combination of a basis for that subspace.
In this case, that subspace is a line, that is a subspace with dimension $n-r=5-4=1$, and we can also use any other vector of that subspace to represent the infinitely many solutions, as for example
$$\lambda(4,2,0,0)$$
or any other multiple vector $\neq 0$.