The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.

Theorem: The general solution to a nonhomogeneous system $Ax = b$ is:

$$x = x_p + x_h$$

where:

• $x_p$ is any particular solution to $Ax = b$
• $x_h$ is any solution to the corresponding homogeneous system $Ax = 0$

Proof sketch:

1. Existence: If $x_p$ satisfies $Ax_p = b$ and $x_h$ satisfies $Ax_h = 0$, then $A(x_p + x_h) = Ax_p + Ax_h = b + 0 = b$, so $x_p + x_h$ is a solution.
2. Completeness: Every solution to $Ax = b$ can be written as $x_p + x_h$ for some $x_h$ in the null space. If $x$ is any solution, then $A(x - x_p) = b - b = 0$, so $x - x_p$ is in the null space.

Example: For the system:
$$\begin{cases} x + y = 4 \\ 2x - y = 2 \end{cases}$$

Particular solution: $x_p = (2, 2)$
Homogeneous solution: $x_h = (0, 0)$ (unique in this case)
General solution: $(2, 2)$