Solve x + y + z = 0, x - 2y + z = -6, 2x + y - z = 5 by gaussian…

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The question

Question: solve x + y + z = 0, x − 2y + z = −6 and 2x + y − z = 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 a 1 in row 2

4 Get 0s under the 1 in column 2

5 Get a 1 in row 3

6 Back substitution

Start at the bottom.

Each row is an equation again.

Row 3 is 1z = −7/3, with 0 for the other letters.

So z = −7/3.

Each row is an equation again.

Row 2 is 1y = 2.

So y = 2.

Row 1 is the equation x + y + z = 0.

We know y = 2 and z = −7/3.

We put them in.

Work it out: x − 1/3 = 0.

To remove the − 1/3, we add 1/3 on both sides.

So x = 1/3.

We write it as (1/3, 2, −7/3).

7 Check

Let's check in every equation.

We put x = 1/3, y = 2, z = −7/3 into ①.

We get 0, the same as the right side.

Now ②.

We get −6, the same as the right side.

Now ③.

We get 5, the same as the right side.

All of them work.

So the answer is right!

Let's look back.

First: write the augmented matrix.

Then: get 0s under the 1 in column 1.

Then: get a 1 in row 2.

Then: get 0s under the 1 in column 2.

Then: get a 1 in row 3.

Then: back substitution.

Last: check.