Question

Solve the linear system using the Gauss-Jordan elimination method for the following equations:

-2x + 4y - 18 = 0
2x + 3y - 10 = 0

Answer 1

The Gauss-Jordan elimination method is used to transform the augment matrix of the given linear equations into reduced row-echelon form, resulting in the solution x = 1 and y = -4.

Step-by-step explanation:

To solve the linear system using the Gauss-Jordan elimination method, we first write the system of equations as an augmented matrix:

1. Write the system as an augmented matrix:
   [-2 4 | -18]
   [ 2 3 | -10]
   
2. Use row operations to convert the matrix to reduced row-echelon form. Start by adding the first row to the second row to eliminate the 'x' term in the second row:
   [ -2 4 | -18 ]
   [ 0 7 | -28 ]
   
   This is done by R2 = R2 + R1.
3. Next, we make the leading coefficient of the first row '1' by multiplying the first row by -1/2:
   [ 1 -2 | 9 ]
   [ 0 7 | -28 ]
   
4. After that, we create a '1' as the leading coefficient in the second row by multiplying the second row by 1/7:
   [ 1 -2 | 9 ]
   [ 0 1 | -4 ]
   
5. Now, we take the second row and multiply it by 2, then add it to the first row in order to create a zero above the leading '1' in the second row:
   [ 1 0 | 1 ]
   [ 0 1 | -4 ]
   This is achieved by R1 = R1 + 2*R2.

This results in the solution x = 1 and y = -4.