Linear Equation
AlgebraA 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 that makes a straight line when you graph it. It has a variable like $x$, but $x$ is never squared or raised to any other power.
Example
$2x + 3 = 11$ is a linear equation. Solve it by subtracting $3$ from both sides to get $2x = 8$, then divide by $2$ to get $x = 4$.
Key Insight
The word "linear" comes from "line." If the equation would draw a straight line on a graph, it is linear.
Definition
A linear equation in one variable 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 a straight line.
Example
$3x - 7 = 2$: add $7$ to both sides to get $3x = 9$, then divide by $3$ to get $x = 3$. In two variables, $y = 2x + 1$ is satisfied by $(0,1)$, $(1,3)$, $(2,5)$, all collinear.
Key Insight
A linear equation has at most one solution in one variable. In two variables it has infinitely many solutions, all forming a straight line on the coordinate plane.
Definition
A linear equation is a polynomial equation of degree $1$. In $n$ variables it defines a hyperplane in $\mathbb{R}^n$. The general form $a_1x_1 + a_2x_2 + \ldots + a_nx_n = b$ represents an affine subspace of dimension $n-1$. Linear equations are the building blocks of linear systems, whose solution theory is studied via matrix algebra and Gaussian elimination.
Example
The equation $2x - 5y = 10$ defines a line in $\mathbb{R}^2$ with slope $2/5$ and y-intercept $-2$. In matrix form $Ax = b$ with $A = [2, -5]$, $x = [x; y]$, $b = [10]$.
Key Insight
Linearity means the equation respects superposition: if $x_1$ and $x_2$ are solutions to the homogeneous form $ax + by = 0$, so is any linear combination. This property underpins the entire theory of linear algebra.