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Exploring the fundamentals of linear equation systems, this overview covers the methods for solving them, including graphical, substitution, elimination, and matrix techniques. It delves into the consistency and types of solutions, highlighting the practical applications in business, environmental science, and beyond. Homogeneous systems and their significance in fields like physics and computer graphics are also discussed.

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## Definition and Purpose

### Linear Equations

Linear equations are mathematical expressions that involve variables raised to the first power

### System of Linear Equations

A system of linear equations is a collection of linear equations with shared variables, with the goal of finding values that satisfy all equations concurrently

### Purpose of Solving Systems

The purpose of solving systems of linear equations is to determine the values of variables that satisfy all equations in the system

## Types of Solutions

### Unique Solutions

Unique solutions occur when a system has exactly one solution, indicating it is independent

### Infinite Solutions

Infinite solutions occur when a system has infinitely many solutions, indicating it is dependent

### No Solution

No solution occurs when a system has parallel lines, indicating there is no solution, or coinciding lines, indicating infinitely many solutions

## Methods of Solving

### Graphical Solutions

Graphical solutions involve visually representing the equations on a coordinate plane to find the intersection points, which represent the solutions

### Algebraic Methods

Algebraic methods, such as substitution and elimination, involve manipulating equations to isolate and solve for variables

### Matrix Techniques

Matrix techniques, such as Gaussian elimination and Row Echelon Form, are efficient and systematic methods for solving larger systems of linear equations

## Applications

### Business

Systems of linear equations are used in business to optimize profit by solving linear inequalities that represent constraints

### Environmental Science

Environmental scientists use systems of linear equations to model and predict pollution levels in ecosystems

### Physics and Computer Graphics

Homogeneous systems of linear equations are particularly relevant in physics for analyzing equilibrium states and in computer graphics for transformations