Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. ﻿{w+x−y+z=−22w−x+2y−z=7−w+2x+y+2z=−1\begin{cases}
w + x - y + z = -2 \
2w - x + 2y - z = 7 \
-w + 2x + y + 2z = -1
\end{cases}⎩⎨⎧​w+x−y+z=−22w−x+2y−z=7−w+2x+y+2z=−1​

Accepted Answer

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

$$\begin{cases} w + x - y + z = -2 \ 2w - x + 2y - z = 7 \ -w + 2x + y + 2z = -1 \end{cases}$$

The augmented matrix is:

$$\left[ \begin{array}{cccc|c} 1 & 1 & -1 & 1 & -2 \ 2 & -1 & 2 & -1 & 7 \ -1 & 2 & 1 & 2 & -1 \end{array} ight]$$ Use row operations to create zeros below the leading 1 in the first column (pivot position). Specifically:
- Replace Row 2 with Row 2 minus 2 times Row 1.
- Replace Row 3 with Row 3 plus Row 1.

This will help in forming an upper triangular matrix. Next, focus on the second row and second column to create a leading 1 (if necessary) and then eliminate the entry below it in the third row. Use appropriate row operations such as scaling and adding multiples of rows to each other. After obtaining an upper triangular matrix, use back substitution to express variables in terms of each other or constants. This involves solving the last equation for one variable, then substituting back into the previous equations. Write the solution set clearly, indicating if there are free variables (parameters) or a unique solution. If any inconsistency arises (like a row with all zeros except the augmented part), conclude that no solution exists.

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 + x - y + z = -2 \\2w - x + 2y - z = 7 \\-w + 2x + y + 2z = -1\(\end{cases}$

Verified video answer for a similar problem:

Key Concepts

Systems of Linear Equations

Gaussian Elimination

Row Operations and Consistency