A linear system is a collection of linear equations whose unknowns must satisfy every equation at the same time.

For example,

$$\begin{aligned} x+y&=5,\ 2x-y&=1 \end{aligned}$$

is a system in the unknowns $x$ and $y$. Solving the system means finding all pairs $(x,y)$ that make both equations true.

Equations as simultaneous constraints

Each equation restricts the possible values of the unknowns. In two variables, each linear equation represents a line. A solution of the system is therefore a point lying on every line in the system.

For the example above, adding the equations gives

$$3x=6,$$

so $x=2$ and then $y=3$. Thus the system has the solution

$$(x,y)=(2,3).$$

Solution sets

A linear system can have one solution, many solutions or no solution at all. The goal is always to describe the complete solution set, not merely to find one pair that happens to work.

Homogeneous systems

A system is homogeneous when every constant term is zero, for example

$$\begin{aligned} x+2y&=0,\ 3x-y&=0. \end{aligned}$$

The all-zero solution is always present in a homogeneous system. Nonzero solutions may also exist.

Linear systems turn several simultaneous linear relationships into a single problem that can later be handled systematically with matrices.