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A system of linear equations is a collection of one or more linear equations involving the same set of variables. In this particular exercise, we are working with a set of three linear equations that involve the variables \(x\), \(y\), and \(z\). These equations are:

• \(3x - 9y + 6z = -12\)
• \(x - 3y + 2z = -4\)
• \(2x - 6y + 4z = 8\)

The goal with systems like these is to find values for each of the variables that satisfy all the equations simultaneously. There are several methods to solve such systems, and in this case, we're utilizing the Gauss-Jordan elimination method. This method helps simplify these equations into a form that makes the solutions more apparent.

To solve a system of linear equations more systematically, we often convert it into an augmented matrix. This matrix combines all coefficients of the variables, with an additional column that represents the constants from the right-hand side of the equations. For our system, the augmented matrix looks like this:\[\begin{array}{ccc|c}3 & -9 & 6 & -12 \1 & -3 & 2 & -4 \2 & -6 & 4 & 8 \\end{array}\]The first three columns correspond to the coefficients of \(x\), \(y\), and \(z\), and the last column contains the constants from the equations. Using this matrix form facilitates the application of row operations, which are crucial for simplifying the system into a more easily solvable form.

Row operations are techniques used in linear algebra to manipulate rows of a matrix. These operations include swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting the multiples of one row from another. In the Gauss-Jordan elimination process, row operations are employed to simplify the augmented matrix into reduced row-echelon form (RREF). This process involves steps like:

• Getting a '1' in the pivot position 鈥� typically, the top-left position of the matrix.
• Making zeros below and above this pivot for simpler variables isolation.

Through steps such as dividing Row 1 by 3, subtracting Row 1 from Row 2, and subtracting 2 times Row 1 from Row 3, we systematically transform the matrix to assess the presence of solutions.

An inconsistent system of equations is one where there are no solutions. This typically occurs when the matrix operations yield a row in the form \([0 \, 0 \, 0 | c]\) where \(c\) is a non-zero constant. This form implies that the equations present a contradiction, such as stating "0 equals a non-zero number."In our exercise, the last row of the matrix simplifies to \([0 \, 0 \, 0 | 16]\). This indicates that no combination of \(x\), \(y\), and \(z\) can satisfy all the equations simultaneously, confirming that the system is inconsistent. In real-world applications, this kind of result might indicate parallel lines that never intersect or other scenarios without a logical solution.