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When dealing with systems of equations, an essential step is to convert the system into an augmented matrix. This matrix includes both the coefficients of the variables and the constants from the equations. For example, in the given system, we have three equations with variables \(x\), \(y\), and \(z\). Each equation contributes a row of numbers to the matrix. The coefficients of \(x\), \(y\), and \(z\) become the entries of the matrix, alongside the vertical bar which separates the coefficients from the constants on the right side of each equation.

This transformation is crucial because it prepares the equations for further manipulations using matrix techniques. The augmented matrix allows for a structured approach to solve the system by applying row operations and ultimately using methods like Gaussian elimination.

Row operations are a powerful tool used to simplify an augmented matrix into a form that is easier to solve. There are three types of row operations you can perform:

• Swap two rows.
• Multiply a row by a non-zero constant.
• Add or subtract a multiple of one row to another row.

Using these operations, our goal is to obtain an upper triangular matrix, where all entries below the diagonal are zeros. This form makes the process of back substitution straightforward. For instance, in the solution provided, we performed operations like multiplying a row and adding it to another to eliminate the coefficients below the main diagonal. This systematic approach is what makes Gaussian elimination such an efficient method.

After converting an augmented matrix to row-echelon form (upper triangular form), the next step is back substitution. This technique solves for the variables starting with the last row and working upward.

In our case, once the matrix is in an upper triangular form, we can easily find the value of \(z\) from the last row. With \(z\) known, we substitute it back into the previous rows to solve for \(y\) and then \(x\). Each step involves simple arithmetic to isolate the variable. It's like unraveling a mystery, step by step, until all variables are uncovered and the solution is fully revealed.

A system of equations, simply put, is a set of equations with multiple variables. The aim is to find values for these variables that satisfy all the equations simultaneously. Systems can be solved in various ways, including graphical, substitution, elimination methods, or matrix methods such as Gaussian elimination.

Understanding systems of equations is fundamental because they are widely applicable in various fields such as engineering, physics, and economics. Solving such systems efficiently can provide insights into complex problems where multiple factors interact. Through methods like Gaussian elimination, we not only find solutions but also gain deeper understanding of how mathematical interactions unfold.