Discuss the Gaussian elimination method for solving systems of linear equations. Provide an example.

The Gaussian elimination method is a widely used technique for solving systems of linear equations. It involves transforming the system into an equivalent triangular system, which can then be easily solved by back substitution.

The method consists of the following steps:

1. Write the augmented matrix of the system, where the coefficients of the variables and the constants are arranged in a rectangular array.

2. Perform row operations to transform the matrix into an upper triangular form. The row operations include multiplying a row by a nonzero constant, adding or subtracting a multiple of one row from another row, and interchanging two rows.

3. Starting from the bottom row, solve for the variables one by one using back substitution. Substitute the values of the variables already solved into the equations above to find the remaining variables.

4. Verify the solution by substituting the obtained values back into the original system of equations.

Let's consider an example to illustrate the Gaussian elimination method:

Consider the following system of linear equations:
2x + 3y - z = 1
4x - y + 2z = -2
x + 2y + 3z = 3

Step 1: Write the augmented matrix:
[2 3 -1 | 1]
[4 -1 2 | -2]
[1 2 3 | 3]

Step 2: Perform row operations to transform the matrix into an upper triangular form:
R2 = R2 - 2R1
R3 = R3 - (1/2)R1

[2 3 -1 | 1]
[0 -7 4 | -4]
[0 1 3.5 | 1.5]

R3 = R3 + (1/7)R2

[2 3 -1 | 1]
[0 -7 4 | -4]
[0 0 4.5 | -0.5]

Step 3: Solve for the variables using back substitution:
z = (-0.5) / 4.5 = -1/9
y = (-4 - 4z) / -7 = 4/7
x = (1 - 3y + z) / 2 = 1/2

Step 4: Verify the solution:
Substituting the obtained values back into the original system of equations:
2(1/2) + 3(4/7) - (-1/9) = 1
4(1/2) - (4/7) + 2(-1/9) = -2
(1/2) + 2(4/7) + 3(-1/9) = 3

All the equations are satisfied, confirming that the solution x = 1/2, y = 4/7, z = -1/9 is correct.

In conclusion, the Gaussian elimination method is an efficient technique for solving systems of linear equations by transforming the system into an equivalent triangular form and then solving for the variables using back substitution.