System of Equations
AlgebraA 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 the system means finding the values of the variables that make all equations true at the same time.
Example
System: $x + y = 10$ and $x - y = 2$. The solution $x = 6$, $y = 4$ satisfies both: $6 + 4 = 10$ and $6 - 4 = 2$.
Key Insight
Think of two clues in a mystery. Each equation is a clue, and the solution is the only answer that fits every clue at once.
Definition
A system of linear equations in two variables consists of two (or more) equations, each defining a line. The solution is the set of ordered pairs $(x, y)$ satisfying all equations. Systems can be solved by substitution, elimination, or graphing. Solutions can be one point, no points (parallel lines), or infinitely many (same line).
Example
System: $2x + y = 7$ and $x - y = 2$. Add the equations: $3x = 9$, so $x = 3$. Substitute: $3 - y = 2$, $y = 1$. Solution: $(3, 1)$.
Key Insight
The three solution types correspond to three geometric situations: lines intersecting (one solution), parallel lines (no solution), and overlapping lines (infinitely many solutions).
Definition
A linear system of $m$ equations in $n$ variables can be written as $Ax = b$, where $A$ is an $m \times n$ matrix, $x$ is the $n$-dimensional variable vector, and $b$ is the $m$-dimensional constant vector. By the Rouche-Capelli theorem, the system is consistent iff $\text{rank}(A) = \text{rank}(A|b)$. If consistent, the solution is unique iff $\text{rank}(A) = n$; otherwise there are infinitely many solutions parameterized by $n - \text{rank}(A)$ free variables.
Example
Two equations in two unknowns: $A$ is $2 \times 2$. If $\det(A)$ is not zero, unique solution $x = A^{-1}b$. If $\det(A) = 0$, either no solution or infinitely many.
Key Insight
Systems of equations are the foundation of linear algebra. Every numerical algorithm in science and engineering (finite element analysis, machine learning, circuit simulation) reduces ultimately to solving large linear systems.