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In the realm of solving systems of linear equations, an augmented matrix plays a crucial role. An augmented matrix combines the coefficients of the variables from each equation in the system with the constants from the other side of the equation.
This combination creates a numerical snapshot of the entire system.To construct an augmented matrix:

• List the coefficients of each variable from all equations in a tabular format.
• Include a vertical bar to demarcate the coefficients from the constants.
• Place the constants from each equation on the other side of the vertical line.

For instance, given the system of equations:\[x+2y-z=1 \-x-2y+2z=-2 \3x+6y-3z=5\]the augmented matrix is structured as follows:\[\begin{bmatrix} 1 & 2 & -1 & | & 1 \-1 & -2 & 2 & | & -2 \3 & 6 & -3 & | & 5\end{bmatrix}\]This compact representation is beneficial for applying techniques like Gaussian elimination to solve systems efficiently.

A system of linear equations consists of multiple linear equations working together. Each equation within the system shares common variables. The goal is to find the values for these variables that satisfy all the equations simultaneously.
Linear equations are simpler because they don't involve powers or roots of variables. This makes the analysis and solution more straightforward using methods like substitution, elimination, or matrices.Key features of a linear system include:

• Each term is either a constant or the product of a constant and a single variable.
• The equations can be two-dimensional (involving 'x' and 'y') or higher depending on the number of variables.
• Graphically, they represent straight lines when plotted on a graph.

In our given system, the equations form a set:\[\begin{array}{c} x+2 y-z=1 \-x-2 y+2 z=-2 \3 x+6 y-3 z=5 \end{array}\]The solution to this set reveals a point or set of points where all equations intersect. However, sometimes the system can be inconsistent, showing no common solution.

An inconsistent system of linear equations is one where no possible solution satisfies all equations simultaneously. This typically arises when one or more of the equations contradicts the others.
To identify inconsistency, examine the reduced form of the augmented matrix after performing Gaussian elimination. If a row in the matrix results in an equation like:\[0x + 0y + 0z = c\]where \(c\) is a non-zero constant, it indicates inconsistency. This is because zero can never equal a non-zero number, making the system unsolvable.In the original exercise, after working through the equations:\[\begin{bmatrix} 1 & 2 & -1 & | & 1 \0 & 0 & 1 & | & -1 \0 & 0 & 0 & | & 2\end{bmatrix}\]the third row became\[0x + 0y + 0z = 2\],which is impossible since zero times any number will not equal 2. Thus, the system is labeled as inconsistent, indicating that no solution exists that satisfies all equations.