Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. ﻿{w−2x−y−3z=−9w+x−y=03w+4x+z=62x−2y+z=3\begin{cases}
w - 2x - y - 3z = -9 \
w + x - y = 0 \
3w + 4x + z = 6 \
2x - 2y + z = 3
\end{cases}⎩⎨⎧​w−2x−y−3z=−9w+x−y=03w+4x+z=62x−2y+z=3​

Accepted Answer

Write the system of equations as an augmented matrix. The system is:

$$\begin{cases} w - 2x - y - 3z = -9 \ w + x - y + 0z = 0 \ 3w + 4x + 0y + z = 6 \ 0w + 2x - 2y + z = 3 \end{cases}$$

The augmented matrix is:

$$\left[\begin{array}{cccc|c} 1 & -2 & -1 & -3 & -9 \ 1 & 1 & -1 & 0 & 0 \ 3 & 4 & 0 & 1 & 6 \ 0 & 2 & -2 & 1 & 3 \end{array}ight]$$ Use row operations to create zeros below the leading 1 in the first column (pivot position). For example, subtract Row 1 from Row 2 to eliminate the first element in Row 2, and subtract 3 times Row 1 from Row 3 to eliminate the first element in Row 3. Next, move to the second column and create a leading 1 in the second row, second column if necessary. Then use row operations to create zeros above and below this pivot. This involves manipulating Rows 3 and 4 to eliminate the second column entries below and above the pivot. Continue this process for the third and fourth columns, creating leading 1s (pivots) and zeros in all other positions of those columns. This will transform the matrix into reduced row echelon form (RREF). Once the matrix is in RREF, translate the matrix back into equations. From these equations, express the variables in terms of any free variables if they exist, or find the unique solution if the system is consistent and determined.

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases}\)w - 2x - y - 3z = -9 \(\w\) + x - y = 0 \\3w + 4x + z = 6 \\2x - 2y + z = 3\(\end{cases}$

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Key Concepts

Systems of Linear Equations

Gaussian Elimination

Row Operations and Consistency of Systems