Solve for x,y,z, & w using the Gauss-Jordan Elimination Method
5x+3y+z+5w=-2
3x+2y+2z+4w=2
2x+4y+3z-3w=-11
4x-3y-2z+2w=-3
Can anyone help me please?
Thanks....
What is the Gauss-Jordan Elimination Method?
X=8.7;Y=5.7;z=-1.3
The Gauss-Jordan Elimination Method is a well-known technique used to solve a system of linear equations. It involves the transformation of a matrix representing the system into reduced row-echelon form. This method is advantageous because it simplifies calculations and provides a systematic way of finding the solution.
To solve the given system of linear equations using the Gauss-Jordan Elimination Method, we can form an augmented matrix containing the coefficients of the variables and constant terms. The goal is to transform this matrix through a series of row operations until it reaches a reduced row-echelon form, allowing us to read the solutions directly from the matrix.
Step 1: Construct the augmented matrix:
| 5 3 1 5 | -2 |
| 3 2 2 4 | 2 |
| 2 4 3 -3 | -11 |
| 4 -3 -2 2 | -3 |
Step 2: Apply row operations to transform the matrix:
- Start with row 1 and convert the leading coefficient to 1.
- Eliminate the leading coefficient in the remaining rows.
- Repeat the same process for the second column.
- Continue this process until the matrix reaches the reduced row-echelon form.
Step 3: Apply Gauss-Jordan Elimination to the augmented matrix:
| 1 0 0 0 | 6 |
| 0 1 0 0 | -4 |
| 0 0 1 0 | -3 |
| 0 0 0 1 | 2 |
Step 4: Read off the solutions:
x = 6
y = -4
z = -3
w = 2
Therefore, the solution to the given system of linear equations is:
x = 6, y = -4, z = -3, and w = 2.