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The row-reduced form of a matrix, also known as row-reduced echelon form (RREF), is crucial in solving systems of linear equations. It simplifies the matrix associated with the system so that we can easily read off the solutions or determine characteristics of the system, such as consistency and the number of solutions.

For a matrix to be in RREF, it must satisfy the following conditions:

• Any zero rows are at the bottom of the matrix.
• The leading entry (first non-zero number from the left) in a non-zero row is 1, known as a pivot.
• This pivot is to the right of the pivot in the row above it.
• All entries directly above and below a pivot are zeros.
• If a column contains a pivot, every other entry in that column is zero.

Understanding row-reduced matrices enables us to quickly move forward and determine whether a system has one solution, no solution, or infinitely many solutions.

An inconsistent system of equations is one that has no solution. This occurs when the equations represent parallel lines that never intersect, which, in matrix terms, is indicated by a row of all zeroes except for the last entry in an augmented matrix.

During the process of row reduction, if we encounter a row that has the form \(\left[\begin{array}{ccc|c}0 & \cdots & 0 & k\end{array}\right]\) where \(k\) is not zero, this row tells us that the system is inconsistent because it suggests an impossible solution, such as \(0x_1 + 0x_2 + \ldots = k\) where \(k\) is non-zero. It's like saying 'nothing equals something,' which is clearly a contradiction.

A system of equations has infinitely many solutions when there are more unknowns than there are independent equations. In these cases, some variables do not have a unique value and are called free variables.

A telltale sign of this in a row-reduced matrix is the presence of at least one non-pivot column (a column without a leading one), indicating that the corresponding variable can take on multiple values. When you have free variables, you can express the basic (pivot) variables in terms of the free variables, leading to an infinite set of solutions or a solution space, typically described with parameterized variables.

In the context of solving linear systems, basic variables correspond to columns with pivots in the row-reduced matrix; these variables are dependent on the other variables and can be directly solved for. Free variables are associated with non-pivot columns and represent the set of variables that can be assigned arbitrary values; their values are 'free' to be chosen.

In the step-by-step solution of our exercise, \(x1, x2, \text{ and } x3\) are the basic variables, determined by the columns with pivots in the row-reduced matrix. Variable \(x4\) is a free variable, as its column in the matrix lacks a pivot. This means that \(x4\) can take any real value, and we can express \(x3\) in terms of \(x4\) to give us a family of solutions. The system has infinitely many solutions because there's an entire dimension of values that \(x4\) can take, each leading to a different set of values for the basic variables.