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# Definition of system of linear equations

Put content here**Definition:** A **solution** to a system of linear equations is an ordered $n$-tuple $(s_1, s_2, \ldots, s_n)$ that satisfies **every** equation in the system simultaneously.
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That is, when each variable $x_i$ is replaced by $s_i$, all equations in the system become true equalities.
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**Example:** For the system:
$$\begin{cases}\n2x + y = 5 \\\nx - 3y = -1\n\end{cases}$$
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$(2, 1)$ is a solution because:
- $2(2) + 1 = 5$ ✓
- $2 - 3(1) = -1$ ✓
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$(1, 3)$ is **not** a solution because $2(1) + 3 = 5$ ✓ but $1 - 3(3) = -8 \neq -1$ ✗.
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A system may have zero solutions (inconsistent), exactly one solution (unique), or infinitely many solutions.

# Parents

* Basic terminology