Linear Equations

A linear equation in $n$ variables $x_1, x_2, \ldots, x_n$ is an equation that can be written in the form:

where $a_1, a_2, \ldots, a_n$ and $b$ are constants, and at least one coefficient $a_i$ is non-zero.

Geometric Interpretation

• A linear equation in two variables $x$ and $y$ represents a line in the 2D plane.
• A linear equation in three variables $x$, $y$, and $z$ represents a plane in the 3D space.
• A linear equation in more variables represents a hyperplane.

Systems of Linear Equations

A system of linear equations is a collection of linear equations. For example:

A solution to a system is an assignment of values to the variables that satisfies all equations simultaneously.

Geometrically, they corresponds to intersection of the geometric objects represented by each equation. For example, the solution $(5, 5)$ to the above system is the point where these two lines intersect:

Number of Solutions

A linear system is called inconsistent if it has no solutions; otherwise, it is called consistent. A consistent linear system has either a unique solution or infinitely many solutions.