# 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?

7 years ago

7 months ago

## 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.