21.6.1. Definition

A system of linear equations is a collection of one or more linear equations involving the same set of variables.

A system of linear equations consists of multiple equations that share the same variables. For example, if we have two equations with the variables x and y, then both equations are part of the same system. The goal in solving such systems is often to find values for the variables that satisfy all the equations simultaneously.

Think of it like coordinating a meeting with friends. Each friend has specific times they are available, represented as a linear equation. To find a suitable time for everyone, you need to find a common time slot that works for all, just like finding variable values that meet all the equations in the system.

Forms: • General Form (2 variables): $a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2$

• Matrix Form: AX =B • where A is the coefficient matrix, X is the variable matrix, B is the constant matrix.

Linear equations can be represented in different forms. The general form shows how a linear relationship can be expressed with coefficients and constants. In matrix form, the equations can be condensed into a structured format where A represents the coefficients of the variables, X represents the variables themselves, and B is a matrix of constants. This matrix form simplifies many calculations, especially when using computational methods to solve the equations.

Imagine you have a recipe that needs specific amounts of ingredients. The general form of your recipe represents the ingredients and their quantities, while the matrix form organizes these ingredients into a shopping list format, making it easier to manage and compute.

Solution Methods • Graphical Method (only practical for 2 or 3 variables) • Substitution and Elimination • Matrix Methods (preferred for large systems): – Gauss Elimination – Gauss-Jordan Elimination – LU Decomposition – Matrix Inversion Method

There are various methods to solve systems of linear equations. The graphical method provides a visual solution for smaller systems (2 or 3 variables) by plotting the equations on a graph. Substitution and elimination are algebraic methods used to simplify the equations and find the variable values. For larger systems, matrix methods, such as Gauss elimination, are preferred because they are systematic and efficient for solving multiple equations simultaneously.

Solving a problem can be approached in different ways. For instance, if you're trying to decide what movie to watch with friends, you could either discuss options verbally (substitution), map out preferences on a chart (graphical), or check a streaming service's recommendations for a larger selection (matrix methods). Each method has its advantages depending on the situation.

• Consistent: At least one solution exists. • Inconsistent: No solution exists. • Infinitely many solutions: When the rank of the augmented matrix equals the number of variables and the system is dependent.

The concept of consistency in a system of linear equations determines whether or not the equations can be solved. A consistent system has at least one solution, meaning the equations do not contradict each other. An inconsistent system has no solutions, indicating that the equations are contradictory. Some systems may have infinitely many solutions, typically when the equations are dependent, meaning they represent the same line or plane in a higher dimension.

Consider a group project with conflicting ideas on how to proceed. If some members suggest conflicting plans (inconsistent), you won't reach an outcome. If everyone agrees on a single plan, you have one solution (consistent). Alternatively, if multiple plans can work and everyone is okay with them, you have many options (infinitely many solutions).

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