Linear Equation
A linear equation is an algebraic equation whose graph is a straight line, containing variables raised only to the first power.
Formula
ax + b = c
Definition
A linear equation is an equation whose graph is a straight line, built from a variable like $x$ that is never squared or raised to any other power. In one variable it has the form $ax + b = c$, where $a$, $b$, and $c$ are constants and $a$ is not zero; in two variables the form is $ax + by = c$, and every solution is an ordered pair $(x, y)$ lying on that line. More formally, a linear equation is a polynomial equation of degree $1$; in $n$ variables the general form $a_1x_1 + a_2x_2 + \ldots + a_nx_n = b$ defines a hyperplane, an affine subspace of dimension $n-1$ in $\mathbb{R}^n$. Linear equations are the building blocks of linear systems, whose solution theory is studied through matrix algebra and Gaussian elimination.
Example
For example, $2x + 3 = 11$ solves by subtracting $3$ from both sides to get $2x = 8$, then dividing by $2$ to get $x = 4$; similarly $3x - 7 = 2$ gives $3x = 9$, so $x = 3$. In two variables, $y = 2x + 1$ is satisfied by $(0,1)$, $(1,3)$, $(2,5)$, all collinear points on the same line. The equation $2x - 5y = 10$ defines a line in $\mathbb{R}^2$ with slope $2/5$ and y-intercept $-2$, which can be written in matrix form $Ax = b$ with $A = [2, -5]$, $x = [x; y]$, $b = [10]$.
Key Insight
The word "linear" comes from "line": if an equation graphs as a straight line, it is linear. A linear equation in one variable has at most one solution, while in two variables it has infinitely many, all forming a line on the coordinate plane. This behavior reflects a deeper property called superposition: if $x_1$ and $x_2$ are solutions to the homogeneous form $ax + by = 0$, so is any linear combination of them, a principle that underpins the entire theory of linear algebra.