System of Linear Equations¶

Definition¶

A system of equations is an ordered collection of equations. A system of linear equations, or linear system, is a system whose equations can all be written in linear form.

The standard form of a linear system with $m$ equations and $n$ unknowns is

\begin{aligned} a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n&=b_1,\\ a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n&=b_2,\\ &\vdots\\ a_{m1}x_1+a_{m2}x_2+\cdots+a_{mn}x_n&=b_m. \end{aligned}

The coefficients form the coefficient matrix $\boldsymbol{A}$ , the unknowns form the vector $\boldsymbol{x}$ , and the right-hand sides form the vector $\boldsymbol{b}$ , so the system can be written as $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ $b$ .

Interpretation¶

A linear system packages several linear constraints into one problem: find all vectors $\boldsymbol{x}$ that satisfy every equation at once. Matrix notation separates the structural information $\boldsymbol{A}$ from the unknown vector $\boldsymbol{x}$ and the right-hand side $\boldsymbol{b}$ .

On a grid, the lines x1 + x2 = 3 and x1 - x2 = 1 crossing at the point (2, 1), the solution of the system.

The illustrated system is $x_1+x_2=3$ and $x_1-x_2=1$ $1$ , or $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ with $\boldsymbol{A}=\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ and $\boldsymbol{b}=\begin{pmatrix}3\\1\end{pmatrix}$ $(31)$ . Each equation describes a line, and their intersection $(2,1)$ is the solution.

"Linear" means that each unknown appears only to the first power, multiplied by a constant: $x_1x_2=1$ $1$ , $x_1^2+x_2=3$ and $\sin x_1=0$ are not linear, while $x_1=3-x_2$ is, since it can be rewritten as $x_1+x_2=3$ $3$ . The matrix form can be read in two ways. Row by row, row $i$ of $\boldsymbol{A}$ with $b_i$ is one equation. Column by column, $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ asks whether $\boldsymbol{b}$ is a linear combination $x_1\boldsymbol{a}_1+\cdots+x_n\boldsymbol{a}_n$ of the columns of $\boldsymbol{A}$ .

In OR Modelling, a corner point of a two-variable linear program is found by turning two constraints into equations and solving them together, which is a $2\times2$ linear system. A common slip is to expect infinitely many solutions whenever there are more unknowns than equations: $x_1+x_2+x_3=1$ $1$ , $x_1+x_2+x_3=2$ has three unknowns and no solution at all.

Key Results¶

Over $\mathbb{R}$ or $\mathbb{C}$ , a linear system can have no solution, exactly one solution, or infinitely many solutions.
The system $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ is solved by applying row operations to its augmented matrix.
The system is consistent if and only if the relevant ranks match: $\operatorname{rank}(\boldsymbol{A})=\operatorname{rank}(\boldsymbol{A}\mid \boldsymbol{b})$ $b)$ .

Examples¶

$x+y=10,\;z+3=5$ is a system of equations.
$3x+y-z=-4,\;x+z=0,\;3x-2y=-3$ is a linear system.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Weeks 3–4

Aliases: linear system, simultaneous linear equations Keywords: system of equations, linear equation, coefficient matrix, right-hand side vector, rhs vector Symbols: complex numbers, real numbers

Last updated: 27 October 2026 11:52