Systems of linear equations

Introduction

A system of linear equations (also known as linear system) is a collection of one or more linear equations involving the same variables. For example,

\[ \begin{align*} 3x + 2y - z & = 1 \\ 2x - 2y + 4z & = -2 \\ - x + \frac{1}{2}y - z & = 0 \end{align*} \]

is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by x = 1, y = -2, z = -2, since it makes all three equations valid.

A photograph of a chalkboard with a system of linear equations written on it.

History

The methods for solving systems of linear equations first arose in ancient times, primarily with the introduction of determinants and matrices by Chinese mathematicians. The systematic use of methods to solve linear systems began in Europe during the 16th century.

Types of Linear Systems

Linear systems can be categorized into several types:

1. **Consistent and independent:** These systems have a single unique solution. They are also referred to as non-degenerate systems. 2. **Consistent and dependent:** These systems have infinitely many solutions. They are also referred to as degenerate systems. 3. **Inconsistent:** These systems have no solution.

Methods of Solution

There are several methods to solve a system of linear equations. These include:

1. **Substitution method:** This involves solving one of the equations for one variable in terms of the others, and substituting this into the remaining equations. 2. **Elimination method:** This involves adding or subtracting the equations in order to eliminate one variable, making it easier to solve for the remaining variable(s). 3. **Matrix method:** This involves writing the system of equations as a matrix equation and then solving that equation.

Applications

Systems of linear equations are central to many fields of science and engineering. They are used in physics to model phenomena, in computer graphics to render 3D scenes, in economics to model supply and demand, and in data fitting to model trends.