91Ó°ÊÓ

A system of linear equations consists of two or more linear equations that share a common set of variables. The task is to find values for these variables that satisfy all the equations simultaneously. A solution to a system is an assignment of numbers to variables that makes each equation hold true. For instance, the system given in our exercise has three equations with the same variables: \( x \), \( y \) and \( z \). The goal is to discover whether there are values of \( x \) , \( y \) , and \( z \) that solve all these equations at the same time.

These systems can have a unique solution, no solution, or infinitely many solutions. The method of Gaussian elimination, which we used in our exercise, is a powerful tool for solving such systems by systematically reducing them to a simpler form.

Any system of linear equations can be represented in augmented matrix form. This format consolidates all the coefficients of the variables and the constants from the right-hand side of the equations into a single matrix. In our exercise, we transformed the given system into an augmented matrix which visually pairs every equation's coefficients side by side, followed by the constants in the last column. This compact representation is incredibly helpful for performing operations that will simplify the system.

The augmented matrix for the given problem is:
\[\left[\begin{array}{ccc|c}4 & 1 & -2 & 6 \-1 & -1 & 1 & -2 \3 & 0 & -1 & 5 \end{array}\right]\]
Each row of the matrix corresponds to an equation, and the vertical bar separates the coefficients on the left from the constants on the right.

Elementary row operations are tools that allow us to manipulate the rows of an augmented matrix without altering the solutions of the system. There are three types of elementary row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row.

By using these operations strategically, as in our exercise, we aim to simplify the matrix to a point where we can easily deduce the solution of the system. These operations are essential in Gaussian elimination, preparing us to achieve the row echelon form.

Row echelon form is a stage in Gaussian elimination where the augmented matrix is simplified such that all nonzero rows are above any rows of all zeros, and each leading coefficient (also known as the pivot) of a nonzero row is to the right of the leading coefficient of the row above it. Furthermore, all entries below these pivots are zeros, making the system much simpler to solve.

In the exercise, we aimed to convert the matrix into this form to identify the nature of the system's solution. The key is to create a stair-step pattern, with each stair's 'nose' being the leading 1 in the row, easing our path to find the solutions.

A dependent system of equations is one where the equations are linked to each other, meaning they don’t provide unique information. Instead, they may represent the same geometric line or plane, which results in an infinite number of solutions.

In the context of the Gaussian elimination used in our exercise, when rows of an augmented matrix turn out to be identical, or one is a multiple of another, it indicates dependency. The presence of identical rows after performing row operations, as happened in our exercise, shows there is an infinite number of solutions that satisfy the given equations. This is because two or more equations essentially repeat the same information about the solution set.