Solve x - y + z = 0, 2x + y - z = 15, -3x + y + z = -12 with a matrix

Step by step

The question

Question: solve x − y + z = 0, 2x + y − z = 15 and −3x + y + z = −12 with a matrix.

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.

2 Get 0s 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 in column 2

5 Get a 1 in row 3

6 Get 0s in column 3

7 Read the answer

Each row is an equation again.

Row 1 is 1x + 0y + 0z = 5.

So x = 5.

Row 2: y = 4.

Row 3: z = −1.

We write it as (5, 4, −1).

8 Check

Let's check in every equation.

We put x = 5, y = 4, z = −1 into ①.

We get 0, the same as the right side.

Now ②.

We get 15, the same as the right side.

Now ③.

We get −12, 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 in column 1.

Then: get a 1 in row 2.

Then: get 0s in column 2.

Then: get a 1 in row 3.

Then: get 0s in column 3.

Then: read the answer.

Last: check.