Explore the fundamentals of linear equation systems, integral to various fields like engineering and economics. Learn about simple and general forms, solution methods like elimination and row reduction, and matrix representations. Understand the characteristics of these systems, including independence, consistency, and equivalence, and delve into the specifics of homogeneous linear systems and their solutions.
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Linear equation systems consist of two or more linear equations with the same variables
Set of variable values
A solution to a system is a set of variable values that satisfy all equations simultaneously
Example
The system 3x + 2y - z = 1, 2x - 2y + 4z = -2, and -x + 0.5y - z = 0 has a solution x = 1, y = -2, z = -2
The term "system" implies that the equations are interconnected and must be considered as a whole
Simple linear systems may consist of a single equation with one variable or a few equations with multiple variables
Coefficients, variables, and constants
The general form of a linear system with m equations and n unknowns is a_i1x_1 + a_i2x_2 + ... + a_inx_n = b_i, where a_ij represents the coefficients, x_j the variables, and b_i the constants
Matrix representation
Linear systems can be efficiently represented using matrices, which is advantageous for computational methods
The number of equations and unknowns in a system determines its solvability
Unique solution
A system with two equations and two variables typically has a unique solution at the point of intersection
No solution
A system with more equations than variables may have no solution
Classification of systems
Systems can be classified as underdetermined, overdetermined, or perfectly determined based on their number of equations and unknowns
A linear system's equations are independent if each provides unique information
A system is consistent if it has at least one solution
Two systems are equivalent if their solution sets are identical, even if the equations are presented differently
The Rouché–Capelli theorem is used to determine the consistency of a system by comparing the ranks of the coefficient matrix and the augmented matrix
00
A ______ of linear equations is made up of two or more equations with the same ______.
system
variables
01
Systems of linear equations are fundamental to ______ algebra and are used in fields like ______, ______, and ______.
linear
engineering
physics
economics
02
For instance, the system 3x + 2y - z = 1, 2x - 2y + 4z = -2, and -x + 0.5y - z = 0 has a solution where x = ______, y = ______, and z = ______.
1
-2
-2
