Using Gaussian Elimination to Solve a 3x3 System of Linear Equations

Gaussian Elimination Overview

The Gaussian Elimination method is used to solve a system of linear equations using a step-by-step process to simplify a matrix until a solution is obtained.
It involves transforming the system into an equivalent one, where it is easier to read the solutions directly.

Performing Gaussian Elimination

Construct augmented matrix: Given a system of three equations, first construct an Augmented Matrix comprising coefficients of variables and constants from the equations. The system ax + by + cz = d is transformed to the augmented matrix [a, b, c

d].

Begin with the first equation. Your goal is to create a leading 1 in the first row. To do this you may have to swap rows or multiply the entire row by a fractional coefficient.
Zero below: Once the leading 1 is established in the first row, perform row operations to get 0’s below this 1.
Second equation: Move to the second row, second column. Just like in the first step, you want to create a leading 1 in the second row, and you may need to perform row operations to achieve this.
Zero below: After creating the leading 1 in the second row, perform row operations to establish 0’s below this 1.
Repeat this process for the third row. You should now have a system where the augmented matrix shows a diagonal of 1’s from the top left to the bottom right, with everything below this diagonal being 0. This is called Row Echelon Form.
At this point, you can solve for the remaining variables in a process called Back-substitution.

Key Rules of Gaussian Elimination

Row swapping - Rows can be swapped to get a desired pivot.
Scale a row - Rows can be multiplied by any non-zero constant.
Row replacement - Subtract multiples of one row from another.

The Back-Substitution Phase

After arranging the matrix into row echelon form, start from the bottommost variable (usually z in a system of three variables), use the row of coefficients corresponding to this variable to replace it in the previous equations.
Repeat this process for the second-to-bottom variable (usually y), and continue until you reach the topmost row.
You should now have the values of all the variables from your original system of equations. These are the solutions to your system.

Note

Remember, Gaussian elimination only works for systems with a single unique solution. Systems with no solutions or infinite solutions require different methods to solve.