System of Equations
A system of equations is a set of two or more equations with the same variables, solved simultaneously to find values that satisfy all equations at once.
Definition
A system of equations is two or more equations that share the same variables; solving it means finding values for the variables that make every equation true at the same time. In two variables, each linear equation defines a line, and the system's solution is the set of ordered pairs $(x, y)$ satisfying all of them; it may be solved by substitution, elimination, or graphing, and can have one solution (lines intersecting), no solution (parallel lines), or infinitely many solutions (the same line). More generally, a system of $m$ equations in $n$ variables can be written as $Ax = b$, and by the Rouche-Capelli theorem the system is consistent if and only if $\text{rank}(A) = \text{rank}(A|b)$; when consistent, the solution is unique if $\text{rank}(A) = n$, otherwise there are infinitely many solutions parameterized by $n - \text{rank}(A)$ free variables.
Example
The system $x + y = 10$ and $x - y = 2$ has solution $x = 6$, $y = 4$, since $6 + 4 = 10$ and $6 - 4 = 2$. For $2x + y = 7$ and $x - y = 2$, adding gives $3x = 9$, so $x = 3$ and $y = 1$. In a $2 \times 2$ matrix system, if $\det(A)$ is not zero there is a unique solution $x = A^{-1}b$; if $\det(A) = 0$, there is either no solution or infinitely many.
Key Insight
Think of two clues in a mystery: each equation is a clue, and the solution is the one answer that fits every clue at once. The three solution types correspond to three geometric pictures, intersecting lines, parallel lines, and overlapping lines, and systems of equations are the foundation of linear algebra: essentially every numerical algorithm in science and engineering, from finite element analysis to machine learning to circuit simulation, ultimately reduces to solving large linear systems.