Systems of Linear Equations

A system of linear equations is a set of two or more linear equations involving the same unknown variables.

The goal of solving a system of linear equations is to find the values of the unknowns that satisfy all the equations at the same time.

What Is a System of Linear Equations?

A system of linear equations contains m equations and n unknowns. The coefficients a_ij and the constant terms b_i are real numbers.

The number of equations and the number of unknowns do not have to be the same. Depending on the problem, a system may contain more equations than unknowns, fewer equations than unknowns, or an equal number of each.

Square Systems. When the number of equations is equal to the number of unknowns (m = n), the system is called a square system.

Understanding the Subscripts

The subscripts i and j indicate the position of each coefficient within the system.

1. Subscript i identifies the equation, or row. It ranges from 1 to m.
2. Subscript j identifies the unknown variable, or column. It ranges from 1 to n.

This notation makes it possible to describe systems of any size in a compact and systematic way.

Solutions of a System of Linear Equations

A solution of a system is an ordered n-tuple of real numbers (s₁, ..., s_n) that satisfies every equation in the system when substituted for the unknowns x₁, ..., x_n.

Depending on the relationships among the equations, a system may have a unique solution, infinitely many solutions, or no solution at all.

The set of all solutions is called the solution set and is usually denoted by the symbol S.

Consistent and Inconsistent Systems

Systems of linear equations are commonly classified into two categories:

• Consistent systems, which have at least one solution.
• Inconsistent systems, which have no solution.

From a geometric point of view, the solution set corresponds to the intersection of the solution sets of the individual equations.

A System with a Unique Solution

The system below has exactly one solution because the two lines intersect at a single point.

The coordinates of the intersection point provide the values of the unknowns that satisfy both equations simultaneously.

A System with No Solution

The following system has no solution because the two lines are parallel and never meet.

Since parallel lines have no points in common, the solution set is the empty set (S = Ø).

A System with Infinitely Many Solutions

In the next example, the two lines are coincident, meaning they lie exactly on top of one another.

Because every point on one line also belongs to the other, every point on the line satisfies both equations. As a result, the system has infinitely many solutions.

This situation occurs when the equations represent the same geometric object, even if they are written in different forms.