Example of solving two linear systems \(Ax=b\) (nonhomogeneous systems) with the same coefficients.

This Linear Algebra video tutorial provides a basic introduction into the Gauss-Jordan elimination which is a process that involves elementary row operations with \(3 \times 3\) matrices which allows you to solve a system of linear equations with \(3\) variables \((x, y, z)\).

❖ Solve two linear systems \(Ax=b_1\) and \(Ax=b_2\) by using a Reduced Row Echelon Form (RREF).

(Sometimes, they called this method as Gauss-Jordan elimination or Gauss-Jordan reduction method).

❖ To solve a linear system of equations by Gauss Jordan elimination, we have to put it in RREF.

So, you need to convert the system of linear equations into an augmented matrix \([A | b_1 | b_2]\) and use matrix row operations to convert the \(3 \times 3\) matrix into the RREF. You can easily determine the answers once you convert them to the RREF.

❖ We have solved the two systems (\(Ax=b_1\) and \(Ax=b_2\)) in the following way:
\([A | b_1 | b_2]\) to \([\text{REFF} | c_1 | c_2]\)

So,
(\(b_1\) and \(b_2\) vectors) changed to (\(c_1\) vector and \(c_2\) vector) because we have done RREF for the augmented matrix \([A | b_1 | b_2]\).