Solve each system. If there is no solution or if there are...

Solve each system. If there is no solution or if there are infinitely many solutions and a system’s equations are dependent, so state.
$$
\left\{\begin{aligned} x-y+3 z &=8 \\ 3 x+y-2 z &=-2 \\ 2 x+4 y+z &=0 \end{aligned}\right.
$$

Key Concepts

System of Linear Equations

This concept refers to a collection of linear equations involving the same set of variables. The goal is to find the values of these variables that satisfy every equation in the system simultaneously. Understanding how to represent and interpret such systems is fundamental in algebra.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by strategically adding or subtracting equations to cancel out one or more variables. This method simplifies the system step by step, reducing it to a form where the values of the remaining variables can be easily determined.

Gaussian Elimination

Gaussian elimination is a systematic procedure for solving systems of linear equations using row operations on an augmented matrix. This process transforms the system into an upper triangular or echelon form from which solutions can be obtained, and it also helps in identifying cases of no solution or infinitely many solutions.

Consistency and Dependency Analysis

In the context of systems of equations, consistency refers to whether there exists at least one set of values that satisfies all the equations. A system is inconsistent if no such set exists, and it is dependent if the equations are multiples of one another, which leads to infinitely many solutions. Analyzing these aspects is crucial for determining the nature of the system's solutions.

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