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A system of linear equations is a collection of two or more linear equations involving the same set of variables. Each equation represents a line, and the solution to the system is the point or points where these lines intersect. Solving such a system means finding the values of the variables that satisfy all equations simultaneously.

For instance, consider two equations in two variables. The system can be solved graphically by plotting each equation on a coordinate plane and identifying the point of intersection. However, this method isn't efficient for complex or higher-dimensional systems. Thus, algebraic methods like substitution, elimination, and matrix approaches, such as Gaussian elimination, are preferred for their ability to handle multiple equations reliably.

Matrix pivoting is a technique used in Gaussian elimination to avoid numerical instability and to increase the accuracy of the solution. Pivoting involves swapping rows or columns in a matrix to move a non-zero element to the pivot position, which is generally on the diagonal.

It ensures that the largest, or a sufficiently large, element is used as the pivot to avoid division by small numbers, which can result in significant rounding errors. In the provided exercise, the element \( -1 \) was chosen as the pivot element, and the goal was to manipulate the other elements in the pivot's column to ease the process of solving the system.

Row operations are the tools used in matrix methods to solve systems of linear equations. They consist of three types of maneuvers: swapping two rows, multiplying a row by a non-zero constant, and adding or subtracting a multiple of one row to another row.

These operations are considered legal moves because they do not change the solution set of the system. Their purpose is to systematically transform the matrix into a simpler form, often aiming for a row-echelon or reduced row-echelon form, from which solutions are easily read off or determined through back-substitution.

Elementary row operations are the building blocks used in row operations. They are fundamental actions we can perform on the rows of a matrix, which correspond to algebraic operations on the equations of a system. The three types include:

• Type 1: Swap the positions of two rows.
• Type 2: Multiply a row by a non-zero scalar.
• Type 3: Add or subtract the multiple of one row to another row.

These operations preserve the equivalence of the original system. In the context of the exercise, an elementary row operation of Type 2 was used to make the pivot element \( -1 \) into \( 1 \) by dividing the first row by \( -1 \) and a Type 3 operation was then used to eliminate the \( 6 \) in the second row to achieve a 0 below the pivot, helping to simplify the system into an upper triangular form.