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Row operations are a set of strategies used in linear algebra that involve manipulating rows of a matrix to simplify solving systems of linear equations. These are critical in transforming matrices, particularly augmented matrices, into more convenient forms. There are three main types of row operations you can perform:

• Swapping two rows: Changing the order of rows can sometimes help simplify calculations. However, the order alone doesn't alter the solution of the system.
• Multiplying a row by a non-zero scalar: This operation will change the coefficients of the equations within the row, making it easier to perform subsequent operations.
• Adding or subtracting a multiple of one row to/from another row: This is probably the most frequently used row operation, aiding significantly in zeroing out elements above or below a pivot to achieve an upper triangular matrix.

In the context of our problem, we applied the row operation on the augmented matrix to simplify it and solve for the variables. Row operations maintain the equality of the system, meaning they will not alter the solution, which makes them powerful tools for solving systems of equations.

A system of equations is essentially a set of two or more equations involving the same set of variables. In this exercise, the system of equations comprised two equations with two variables: \[ x + y = -3 \]\[ 2x - 5y = -6 \]The goal in solving these systems is to find the values of the variables that satisfy all the equations simultaneously. This often involves techniques like substitution, elimination, or using augmented matrices with row operations, as we did here. The advantage of using matrices is that they provide a structured method for solving more complex systems that would be difficult to solve algebraically. Once a system of equations is expressed as an augmented matrix, solving it becomes a straightforward application of row operations to eventually isolate each variable.

An upper triangular matrix is a type of square matrix where all the entries below the diagonal are zero. In the context of solving a system of equations using an augmented matrix, achieving an upper triangular form simplifies the process of solving for the variables.By converting the augmented matrix into an upper triangular form, the system of equations becomes easier to solve using back-substitution. The main advantage is that you can directly find the last variable from the last row, then move upwards row-by-row to find each preceding variable.In this particular problem, we transformed the original matrix to:\[\begin{bmatrix}1 & 1 & \vert & -3 \0 & -7 & \vert & 0\end{bmatrix}\]The second row is an equation only in terms of \( y \), which means once you solve for \( y \), you can substitute it back into the first row to find \( x \). Using the upper triangular matrix form makes solving these equations more systematic and less prone to arithmetic errors.