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The system of linear equations is a collection of one or more equations involving the same set of variables. A linear equation in three variables, such as the ones in the exercise, can be visually interpreted as a plane in three-dimensional space. When we have multiple linear equations, we are looking for a point or points where these planes intersect, which represent the solutions to our system.

For instance, we may have one unique solution where all three planes intersect at a single point, or we may have infinitely many solutions if the planes overlap along a line or coincide completely. There is also a possibility that there is no solution, which occurs if the planes are parallel and never intersect, meaning the system is inconsistent.

The challenge in solving these systems is to find a systematic way to determine whether solutions exist and if they do, what they are. This is where Gaussian elimination, a method used in the provided exercise, comes into play. It transforms the system into an equivalent one that is easier to solve.

To apply Gaussian elimination efficiently, we utilize what is known as an augmented matrix. This compact form of representation includes all the coefficients of the variables along with the constants from the right-hand side of the equations placed into one single matrix. Each row corresponds to an equation and each column, up to the last one, corresponds to a variable's coefficient in the equations.

In the solved exercise, the augmented matrix represents the system using less space and more structure, enabling clear manipulation of the equations. It is formed by aligning the coefficients of the variables in columns and appending the constants on the right, separated by a line to indicate the equality sign from the equations.

Augmented matrices are crucial for visual simplification and for performing row operations that help us reach a solution more systematically. By manipulating rows, which correspond to equations, we work towards simplifying the system without changing its solutions.

The row-reduced form of a matrix, often referred to as row-echelon form, is when it has been simplified through a series of row operations - namely row swapping, multiplying rows by non-zero scalars, and adding or subtracting row multiples to or from other rows.

Our aim with Gaussian elimination is to transform the augmented matrix into this form, where we have a staircase-like pattern with leading coefficients (typically '1's) and zeros beneath them in each column leading up to the last. In the ideal case where we reach a reduced row-echelon form, each leading '1' is the only non-zero entry in its column.

This simplification allows us to interpret the row-reduced matrix as a set of simpler, equivalent equations that can be solved with back-substitution. The process demonstrated in the exercise converts a complex system into one where the solutions to the variables are more apparent and accessible, revealing whether there is a unique solution, infinitely many solutions, or no solution at all.