# Solving Linear Equation Systems

## 1. Introduction

A **linear equation system** is a group of equations (linear) that have more than one **unknown factor**. The unknown factors appear in various equations, but doesn't need to be in all of them. What these equations do is relate all the unknown factors amongst themselves.

*x*and

*y*).

**Solving** this type of problems (a system) consists in finding a value for each unknown factor in a way that it applies to all the equations in the system. The solution of the example system is:

**There isn't always a solution** or there could be **infinite solutions**. If there's only one solution (one value for each unknown factor, like the previous example) we say that the system is **consistent dependent system**. We won't talk about the other kinds because in this section we will only study consistent dependent systems.

## 2. Solving a System

To resolve a system (consistent dependent) **we need at least** the same number of equations as unknown factors. In this section we'll resolve systems (linear) of two equations and two unknown factors with the **methods** we describe next, which are based on obtaining a first degree equation:

**Substitution (elimination of variables):**It consists in isolating one of the unknown factors (for example*x*) and substitute that expression in the other equation. This way we obtain a first degree equation with the unknown factor*y*. Once resolved, we obtain the value of*x*using the value of*y*we know.**Reduction (row reduction):**It consists in operating the equations, for example, adding or subtracting both equations so one of the unknown factors disappears. This way we obtain an equation with only one known factor.**Equalization:**It consists in isolating from both equations the same unknown factor to be able to equal both expression, obtaining one equation with one unknown factor.