Solve the system of equations using the Gauss-Jordan method:
[tex]
\begin{array}{l}
-x+y+z-w=4 \\
3 x-2 y+2 z-3 w=-14 \\
2 x+y-z+w=9 \\
4 x+2 z-3 w=-9 \\
{\left[\begin{array}{cccc|c}
-1 & 1 & 1 & 1 & 4 \\
3 & -2 & 2 & -3 & -14 \\
2 & 1 & -1 & 1 & 9 \\
4 & 0 & 2 & -3 & -9
\end{array}\right]}
\end{array}
[/tex]
Answer (1)
Set up the augmented matrix for the given system of linear equations.
Apply the Gauss-Jordan elimination method to transform the matrix into reduced row echelon form.
Observe that the last row of the reduced matrix leads to a contradiction (0 = 8).
Conclude that the system of equations has no solution. No solution
Explanation
Understanding the Problem We are given a system of 4 linear equations with 4 unknowns: x, y, z, and w. Our goal is to solve this system using the Gauss-Jordan elimination method, which involves transforming the augmented matrix into reduced row echelon form.
Setting up the Augmented Matrix The augmented matrix for the system is: [ − 1 1 1 − 1 4 3 − 2 2 − 3 − 14 2 1 − 1 1 9 4 0 2 − 3 − 9 ] We will perform elementary row operations to transform this matrix into reduced row echelon form.
Applying Gauss-Jordan Elimination After applying the Gauss-Jordan elimination method, the reduced row echelon form of the matrix is: [ 1 0 0 − 0.14285714 0.57142857 0 1 0 0.07142857 2.21428571 0 0 1 − 1.21428571 − 5.64285714 0 0 0 0 8 ]
Interpreting the Result From the last row, we have the equation 0 x + 0 y + 0 z + 0 w = 8 , which simplifies to 0 = 8 . This is a contradiction, indicating that the system of equations has no solution.
Examples
Solving systems of linear equations is crucial in various fields like engineering, economics, and computer science. For instance, in structural engineering, these systems help determine the forces acting on different parts of a bridge. In economics, they can model supply and demand to find equilibrium prices. Understanding how to solve these systems allows engineers and economists to make informed decisions and predictions.
