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.

For example: is an equation system with two equations with two unknown factors (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.

Let's not forget that if we multiply an equation by a number different from 0, the initial equation and the obtained one are equivalent. This means that both equations have the same solutions, therefor, we can work with one or the other. We'll use this property often in the row reduction method.