In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. w + x - y + z = - 2 2w - x + 2y - z = 7 - w + 2x + y + 2z = - 1

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1. Write the system of equations as an augmented matrix: $$\left[\begin{array}{cccccc}1& 1& -1& 1& |& -2\\ 2& -1& 2& -1& |& 7\\ -1& 2& 1& 2& |& -1\end{array}\right]$$
2. Use row operations to get a leading 1 in the first row, first column (if necessary, swap rows or multiply a row by a constant).
3. Eliminate the first column entries below the leading 1 by adding or subtracting multiples of the first row from the other rows.
4. Continue the process to get a leading 1 in the second row, second column, and eliminate the entries below it.
5. Repeat the process for the third row, third column to achieve row-echelon form, then back-substitute to find the solution.

Key Concepts

Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations. It involves transforming the system's augmented matrix into row echelon form using a series of row operations, which simplifies the equations. Once in this form, back substitution can be used to find the values of the variables. This technique is essential for determining whether a unique solution exists, or if the system is inconsistent or has infinitely many solutions.

Row Operations

Row operations are the fundamental manipulations applied to the rows of a matrix during Gaussian elimination. There are three types of row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row from another. These operations help in simplifying the matrix while preserving the solution set of the system of equations, making it easier to analyze the relationships between the variables.

Augmented Matrix

An augmented matrix is a matrix that represents a system of linear equations, including the coefficients of the variables and the constants from the equations. It is formed by appending the column of constants to the coefficient matrix. The augmented matrix is crucial in the Gaussian elimination process, as it allows for a compact representation of the system, facilitating the application of row operations to find solutions or determine the system's consistency.