Step by step
The question
Question: solve x + y + z = 2, x + 2y + z = 3 and 2x + 3y + 2z = 5 by Gaussian elimination.
We want the x, y and z that make all three true.
We will use three moves on the rows.
Swap two rows.
Times a row by a number that is not 0.
Add a multiple of one row to another.
Each move keeps the equations true, like in elimination.
1 Write the augmented matrix
Each row is one equation: the numbers in front of x, y and z, then the number on the right.
The bar stands for the = signs.
This is called an augmented matrix.
We call the rows R 1, R 2 and R 3.
Our goal: 1s down the diagonal, and 0s under them.
This shape is called row-echelon form.
Then we go back up, one letter at a time.
2 Get 0s under the 1 in column 1
Careful!
A row move works on the whole row, the number after the bar too.
That number is the right side of the equation.
3 Get 0s under the 1 in column 2
One row is all 0s.
It says 0 = 0, which is always true.
The equations do not pin down one answer: there are many solutions.
Every point on the line works.
Let's look back.
First: write the augmented matrix.
Then: get 0s under the 1 in column 1.
Last: get 0s under the 1 in column 2.