21.1.1. 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 two or more linear equations that share common variables. This means that the equations are derived from the same contextual need, which often could be solving for the values of those variables that satisfy all equations in the system simultaneously. For example, if we have two equations that need to be solved together, they can be used to find the point where the lines representing those equations intersect.

Imagine trying to find a common meeting point for two friends who start from different locations and head toward a destination. Their paths can be represented by linear equations, and the point where they meet corresponds to the solution of the system of equations.

• 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.

Systems of linear equations can be expressed in several forms. The 'General Form' for systems with two variables shows how they relate through equations. In 'Matrix Form', a system can be represented more compactly, which is especially useful for larger systems. The coefficient matrix A contains the coefficients of the variables, the variable matrix X contains the variables themselves, and the constant matrix B contains the constants from the equations.

Think of a recipe that requires certain amounts of ingredients (variables). You can express the recipe as a list (general form) or as a table (matrix form) that shows how much of each ingredient you need to achieve your cooking goal.

• 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 several methods to solve systems of linear equations. The graphical method involves plotting equations on a graph to find the intersection point visually, which is only feasible for two or three variables. For more complex systems, substitution and elimination methods allow you to manipulate the equations to isolate the variables. Matrix methods like Gauss elimination and matrix inversion are computational approaches that enable solutions for thousands of equations quickly and efficiently, especially suited for computer-led calculations.

Consider if you're trying to balance a budget with income and expenses represented by equations. You can visualize the budget's outcome graphically, or for a more complex budget involving multiple income sources and expenses, you might write down all equations and use substitution or a calculator to adjust and reach a balanced result.

• 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 nature of the solutions available for a system of linear equations is termed consistency. A 'consistent' system has one or more solutions; if it's 'inconsistent', there is no possible set of values for the variables that satisfy every equation in the system. Some systems might allow for an infinite number of solutions, typically when the equations represent the same line—indicating redundancy in information.

Think of a puzzle where some pieces fit together nicely (consistent), while other pieces will never fit with any others (inconsistent). Some puzzles, however, might have multiple ways to combine pieces to achieve the same completed image (infinitely many solutions), illustrating the different scenarios in systems of linear equations.

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