In Problems 17-54, solve each system of equations. If the...

In Problems 17-54, solve each system of equations. If the system has no solution, say that it is inconsisten
$\left\{\begin{array}{l}3 x-2 y+2 z=6 \\ 7 x-3 y+2 z=-1 \\ 2 x-3 y+4 z=0\end{array}\right.$

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all the equations simultaneously. These systems can have a unique solution, infinitely many solutions, or no solution at all, depending on how the equations relate to one another.

Elimination Method

The elimination method involves manipulating the equations in the system in order to eliminate one or more variables, making it easier to solve for the remaining variables. This method typically involves adding or subtracting equations after multiplying them by appropriate constants to cancel out a variable, and is a common approach for solving linear systems.

Consistency of Systems

Consistent systems of equations are those that have at least one solution—either a unique solution or infinitely many solutions. In contrast, an inconsistent system has no solution because the equations contradict one another. Determining the consistency of a system is crucial, as it informs whether further steps in solving the system are meaningful.

Gaussian Elimination and Augmented Matrices

Gaussian elimination is a systematic method for solving systems of linear equations by transforming the corresponding augmented matrix into an upper triangular form, from which the solutions can be easily determined through back-substitution. Using augmented matrices offers a structured approach to performing the elimination method, and helps in identifying the nature of the system—whether it is consistent or inconsistent.

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