Solve each of the following systems. If the solution set is...

Solve each of the following systems. If the solution set is $\varnothing$ or if it contains infinitely many solutions, then so indicate.
$\left(\begin{array}{r}4 x-y+z=5 \\ 3 x+y+2 z=4 \\ x-2 y-z=1\end{array}\right)$

Key Concepts

System of Linear Equations

A system of linear equations is a collection of one 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, which could result in a unique solution, no solution, or infinitely many solutions.

Consistency and Solution Sets

The solution set of a system of equations describes all possible answers that satisfy the equations. A system can be consistent—yielding at least one solution—or inconsistent, meaning there are no solutions. Consistent systems further divide into those having a unique solution or infinitely many solutions, often determined by the number of independent equations relative to the number of variables.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. It involves performing row operations on the augmented matrix of the system to reduce it to an upper triangular form or row echelon form, from which solutions can be easily found, or conclusions can be drawn about the nature of the solution set.

Matrix Representation and Row Operations

Representing a linear system as an augmented matrix allows for the efficient application of row operations such as row swapping, scaling, and linear combinations. These operations help simplify the system while preserving the solution set, serving as the foundation for methods like Gaussian elimination.

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