Gaussian Elimination and Systems of Equations

Systems of Equations

Gaussian elimination is a method used to solve systems of linear equations. It involves manipulating the given system into row echelon form. This process can be performed either by hand or with the help of a computer.

Row Echelon Form

The first step in using Gaussian elimination is finding the row echelon form of a matrix. A matrix is said to be in row echelon form if it satisfies these conditions:

1. Each leading 1 of the matrix must lie directly above another leading 1 of the matrix.
2. If any row has all entries zero except possibly for a single leading 1, then this row may be moved below any other row without changing the solution of the system.
3. Any zero row may precede any nonzero row.

For example, consider the matrix A:

 1  1  | -5    8
 2  3  |  7   10
 4  6  | 15   19

By performing Gaussian elimination, we can transform A into its row echelon form:

 1  1  |  0  -3
 0  0  |  0   0
 0  0  |  0   0

Solving Linear Equations Using Row Reduction

To solve a system of linear equations using Gaussian elimination, follow these steps:

1. Write the system you want to solve in augmented matrix form Ax = b, where A represents the coefficient matrix, x represents the unknown variables, and b represents the constant term.
2. Perform row operations until the augmented matrices are transformed into row echelon form.
3. When you reach the desired row echelon form, the final column will have only two unique elements: zeros and ones. These elements represent the solution(s) to the original system of linear equations.

Example Problem

Consider the following system of equations:

 2x + 3y = 7
 x - y = -2

We can rewrite this system as an augmented matrix:

   2  3  |  7
   1 -1 |  -2

Now perform Gaussian elimination to find the row echelon form:

   2  3  |  7
   0  0  |  0
   0  0  |  0

From the last column, we see that there are multiple solutions:

      x = 7
      y = 2

This means that there are infinitely many pairs (x, y) that satisfy the given equation. For example, (7, 2) and (14, 4) are both valid solutions.