Definition of solution to a system of linear equations

Definition: The solution set of a system of linear equations is the set of all solutions to the system.

For a system $Ax = b$, the solution set is:

$$S = \{x \in \mathbb{R}^n : Ax = b\}$$

The solution set can be:

• Empty ($S = \emptyset$) — the system is inconsistent
• A single point — the system has a unique solution
• Infinite — the system has infinitely many solutions (described parametrically)

Example: For $x + y = 2$, the solution set is the infinite line:
$$S = \{(t, 2 - t) : t \in \mathbb{R}\}$$

For the system $\begin{cases} x + y = 2 \\ x + y = 3 \end{cases}$, the solution set is empty.