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.

Each equation $a_{i1}x_1+a_{i2}x_2=b_i$ is a line; the solutions are the points on both. Make the rows parallel to get no solution or infinitely many.

"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: 20 October 2026 03:02