A system of linear equations consists of two or more linear equations made up of two or more variables, such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all of the system's equations at the same time. Some linear systems may not have a solution, while others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.

In this section, we will focus primarily on systems of linear equations which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables:

$2x + y = 15 \\ 3x - y = 5$

The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. 

$2(4) + 7 = 15 \\ 3(4) - 7 = 5$

Both of these statements are true, so $(4, 7)$ is indeed a solution to the system of equations.

Note that a system of linear equations may contain more than two equations, and more than two variables. For example,

 $3x + 2y - z = 12 \\ x - 2y + 4z = -2 \\ -x + 12y -z = 0 $

is a system of three equations in the three variables $x, y, z$. A solution to the system above is given by

$x = 1 \\ y = -2 \\ z = - 2$

since it makes all three equations valid. 

Types of Linear Systems and Their Solutions

In general, a linear system may behave in any one of three possible ways:

1. The system has a single unique solution.
2. The system has no solution.
3. The system has infinitely many solutions.

Each of these possibilities represents a certain type of system of linear equations in two variables. Each of these can be displayed graphically, as below. Note that a solution to a system of linear equations is any point at which the lines intersect.

Systems of Linear Equations

Graphical representations of the three types of systems.

An independent system has exactly one solution pair $(x, y)$. The point where the two lines intersect is the only solution.

An inconsistent system has no solution. Notice that the two lines are parallel and will never intersect.

A dependent system has infinitely many solutions. The lines are exactly the same, so every coordinate pair on the line is a solution to both equations.