Solve each system using Gaussian elimination. State whether...

Solve each system using Gaussian elimination. State whether each system is independent, dependent, or inconsistent. See the procedure for the Gaussian elimination method for two equations on page 627.

$0.2 x+0.6 y=0.7$
$0.5 x-y=0.5$

Key Concepts

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations. This procedure involves using elementary row operations to reduce a matrix (or set of equations) to an upper triangular or row-echelon form, from which the solutions can be obtained by back-substitution. This method is fundamental in linear algebra as it works for any system regardless of the number of equations and variables.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find values for these variables that satisfy all the equations simultaneously. Understanding the structure of these systems, including their coefficients and constants, is crucial for effective application of solution techniques such as Gaussian elimination.

Consistency and Dependency in Systems

Systems of equations can be classified based on their solution sets. A consistent system has at least one solution, and if it has exactly one unique solution, it is called independent. If there are infinitely many solutions, typically due to one or more equations being redundant, the system is considered dependent. Inconsistent systems, on the other hand, have no solution, typically due to contradictory equations.

Variable Elimination

Variable elimination is a key aspect of Gaussian elimination where one or more variables are systematically removed from the equations. This is done by combining equations so that coefficients of a chosen variable cancel each other out, simplifying the system to make it easier to solve for the remaining variables.

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