Answer :
Gauss-Jordan elimination is a systematic method to solve a system of equations by reducing them to an augmented Explanationmatrix and performing row operations until the matrix is in reduced row-echelon form, from which solutions can be easily read.
To solve the system of equations using Gauss-Jordan elimination, you convert the system into an augmented matrix and then apply a series of row operations to reduce it to reduced row-echelon form (RREF). Here is a step-by-step approach for the given system:
- Write the augmented matrix for the system.
- Perform row operations to get a leading 1 in the first row, first column (pivot position).
- Use the leading 1 to create zeros in the rest of the first column.
- Move to the second row and repeat the process to create a leading 1 and then zeros below it.
- Continue this process for subsequent rows.
- Once you have the matrix in RREF, interpret the last column as the solutions for the variables.
The Gauss-Jordan method transforms a matrix into a form where the left side of the augmented matrix is the identity matrix, and the right side gives the solution for each variable. As the system size increases, Gauss-Jordan elimination can become complex, but it remains more systematic compared to other methods such as substitution.
When variables have been solved, it's important to check by substituting them back into the original equations to confirm that they yield true statements, assuring that the solutions are correct.
