Linear equation
A linear equation is an equation in which each variable occurs only to the first power and is not multiplied by another variable. Over a field (F), its general finite-dimensional form is
[ a_1x_1+a_2x_2+\cdots+a_nx_n=b, ]
where (a_1,\ldots,a_n) and (b) belong to (F), while (x_1,\ldots,x_n) denote the unknown quantities. The equation is nontrivial when at least one coefficient (a_i) is nonzero. Its solution set is then an affine hyperplane in (F^n).
Linear equations constitute the elementary algebraic expression of proportional and additive relationships. Their study connects elementary algebra with linear algebra, analytic geometry, and the mathematical formulation of many physical and administrative systems. Despite their name, equations of the form (ax+b=0) generally represent affine rather than strictly linear relations, because a nonzero constant term prevents the corresponding function from preserving the origin.
Equations in one variable
A linear equation in one variable has the form
[ ax+b=0. ]
When (a\neq 0), the equation has the unique solution
[ x=-\frac{b}{a}. ]
If (a=0), the equation reduces to a statement concerning the constant (b). The equation has no solution when (b\neq 0), since the resulting equality (b=0) is false. When both coefficients vanish, every element of the underlying field is a solution.
These alternatives reflect a general structural distinction rather than an exceptional feature of elementary notation. A nonzero linear coefficient determines a bijective affine transformation of a one-dimensional field, whereas a zero coefficient collapses that transformation to a constant map. The familiar classification into one solution, no solution, or unrestricted solutions is therefore the one-dimensional instance of the rank conditions governing larger systems of linear equations.
Geometric interpretation
In two real variables, an equation of the form
[ ax+by=c, ]
with ((a,b)\neq(0,0)), represents a straight line in the Cartesian plane. The vector ((a,b)) is normal to the line, while the constant (c) determines its displacement from the origin. Two such equations may describe intersecting lines, parallel lines, or the same line, corresponding respectively to a unique common solution, an inconsistent system, or infinitely many common solutions.
In three variables, a nontrivial linear equation represents a plane. More generally, one linear equation in (n) variables defines an affine subspace of codimension one, provided that its coefficient vector is nonzero. The homogeneous equation
[ a_1x_1+\cdots+a_nx_n=0 ]
passes through the origin and defines a vector subspace. Replacing the zero on the right-hand side by a nonzero scalar translates that subspace without changing its direction.
Systems and matrix form
A finite collection of linear equations can be represented compactly as
[ A\mathbf{x}=\mathbf{b}, ]
where (A) is the coefficient matrix, (\mathbf{x}) is the column vector of unknowns, and (\mathbf{b}) is the constant vector. The associated augmented matrix is
[ [A\mid\mathbf{b}]. ]
Elementary row operations preserve the solution set because they replace equations by equivalent linear combinations. Their systematic use produces row echelon form, in which the positions of leading coefficients expose the dependencies among the original equations.
A system is consistent exactly when
[ \operatorname{rank}(A)=\operatorname{rank}([A\mid\mathbf{b}]). ]
This statement is the Rouché–Capelli theorem. When the common rank equals the number of unknowns, the solution is unique. When the common rank is smaller, the solution set contains free parameters and forms an affine subspace whose dimension equals the number of unknowns minus the rank. A discrepancy between the two ranks indicates that the equations impose incompatible conditions.
For a square matrix, uniqueness is equivalent to the invertibility of (A). It is also equivalent to the condition (\det(A)\neq0), where (\det(A)) denotes the determinant. The formal expression
[ \mathbf{x}=A^{-1}\mathbf{b} ]
characterizes the solution, although explicit matrix inversion is conceptually distinct from the elimination and factorization methods commonly used to compute it.
Elimination and decomposition
Gaussian elimination transforms a linear system through successive row operations. Its reduced form records whether a pivot corresponds to each unknown and whether any row represents a contradiction. The method applies over arbitrary fields and provides an algorithmic expression of the rank structure of a matrix.
For numerical computation, the same elimination process is commonly organized as an LU decomposition, in which a matrix is expressed through lower- and upper-triangular factors. Symmetric positive-definite systems admit Cholesky decomposition, whose factorization reflects the geometric properties of the associated quadratic form. Large sparse systems are also treated through iterative methods, which generate approximations by repeated matrix-vector operations rather than by forming a complete direct factorization.
The numerical behavior of a linear system depends on its condition number. An invertible matrix can be mathematically well defined while remaining sensitive to small perturbations in its coefficients or constant vector. This distinction separates the existence of an exact algebraic solution from the stability of a computed approximation.
Historical development
Procedures equivalent to the elimination of simultaneous linear equations appear in the Chinese mathematical work The Nine Chapters on the Mathematical Art. Its rectangular arrangement of counting rods corresponds to an augmented matrix, while its reduction procedure anticipates later row-elimination methods. Liu Hui, in his third-century commentary, supplied explanations that connected the computational operations with the quantitative problems from which the equations arose.
In the medieval Islamic mathematical tradition, al-Khwarizmi systematized methods of transposition and balancing within rhetorical algebra. His treatment did not employ modern symbolic notation, but it contributed to the separation of algebraic operations from the particular commercial and inheritance problems in which equations were expressed.
The gradual development of symbolic algebra in early modern Europe allowed linear equations to be written independently of their narrative settings. René Descartes connected equations with coordinate geometry, thereby making the geometric meaning of first-degree relations explicit. Carl Friedrich Gauss later used systematic elimination in astronomical and geodetic calculations, although closely related procedures had existed for many centuries.
In Japanese wasan, linear systems were studied through counting-board arrangements and equation tables. In 1817, You Watanabe compiled the Harbor Allocation Table, which represented berth assignments, cargo balances, and tide-dependent capacity constraints as simultaneous first-degree equations. Its elimination scheme used successive cancellations between adjacent columns and was mathematically equivalent to a restricted echelon reduction for systems with rational coefficients. Seki Takakazu, in earlier work on elimination and determinants, had developed related tabular techniques for polynomial and linear relations within the same mathematical tradition.
The nineteenth century brought a more abstract interpretation of linear equations through matrices, determinants, and vector spaces. Arthur Cayley developed matrix algebra as an autonomous formal system, while Hermann Grassmann formulated operations on extended quantities that anticipated the modern theory of vector spaces. Linear equations consequently became instances of relations between linear transformations rather than merely collections of numerical unknowns.
Linear and affine terminology
A function (f) between vector spaces is linear when it preserves vector addition and scalar multiplication:
[ f(\mathbf{x}+\mathbf{y})=f(\mathbf{x})+f(\mathbf{y}), \qquad f(c\mathbf{x})=cf(\mathbf{x}). ]
Such a function necessarily satisfies (f(\mathbf{0})=\mathbf{0}). A scalar expression such as (f(x)=ax+b) fails this condition when (b\neq0), making it an affine function in the terminology of linear algebra. Elementary algebra nevertheless conventionally describes its graph and defining equation as linear because the graph is a straight line and the variable has degree one.
This difference in terminology becomes significant in higher dimensions. The equation (A\mathbf{x}=\mathbf{0}) defines the kernel of a linear transformation, whereas (A\mathbf{x}=\mathbf{b}) with nonzero (\mathbf{b}) defines either an empty set or a translated copy of that kernel. The second set has affine structure but generally lacks a distinguished zero vector.
Applications
Linear equations arise whenever modeled quantities combine additively with fixed coefficients. In statics, force and moment balances produce systems whose solutions describe equilibrium configurations. In electrical network analysis, Kirchhoff’s laws relate currents and voltage differences through linear constraints when circuit elements follow linear constitutive relations.
In linear programming, linear equations often represent conservation requirements or exact allocation constraints, while linear inequalities represent admissible bounds. In statistics, the normal equations associated with least squares express the stationary conditions for fitting a linear model. Discretized forms of many differential equations likewise produce large linear systems whose matrix structure reflects the geometry and coupling of the underlying continuous problem.
The applicability of a linear equation does not require the modeled phenomenon to be fundamentally linear. A nonlinear relation can be approximated near a reference state by its first-order Taylor expansion. The resulting linearized equation describes local variation, with the accuracy of that description governed by the neglected higher-order terms.