How to solve systems of linear equations

In mathematics and linear algebra, a system of linear equations, also known as a linear system of equations or simply a linear system, is a set of linear equations (that is, a system of equations in which each equation is first degree).

An example of a linear system of equations would be the following:

Methods of solving systems of linear equations

Solve a system of linear equations is to find all their solutions.

The methods of equalization, substitution and reduction consist of finding and solving, for each of the unknowns, an equation with that unknown and with no other.

Substitution method

The substitution method consists in clearing one of the equations with any unknown, preferably the one with the lowest coefficient, and then substitute it in another equation for its value.

In case of systems with more than two unknowns, the selected one must be replaced by its equivalent value in all the equations except the one we have cleared it. At that moment, we will have a system with an equation and an unknown less than the initial one, in which we can continue applying this method repeatedly.

Equalization method

The equalization method can be understood as a particular case of the substitution method in which the same incognita is solved in two equations and then the right part of both equations are equated with each other.

Reduction method

This method is mostly used in linear systems, with few cases in which it is used to solve non-linear systems. The procedure, designed for systems with two equations and unknowns, consists of transforming one of the equations (generally, by products), so that we obtain two equations in which the same unknown appears with the same coefficient and different sign. Next, both equations are added, thus producing the reduction or cancellation of said unknown, thus obtaining an equation with a single unknown, where the resolution method is simple.