Solve x + 2y + z = 5, x + 3y + 3z = 2, 2x + y - 3z = 17 with a matrix Step by Step

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The question

Question: solve x + 2y + z = 5, x + 3y + 3z = 2 and 2x + y − 3z = 17 with a matrix.

x + 2y + z = 5①

x + 3y + 3z = 2②

We want the x, y and z that make all three true.

2x + y − 3z = 17③

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.

swap two rows

× a row by a number (not 0)

add a multiple of one row to another

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.

= [1, 2, 1, 5; 1, 3, 3, 2; 2, 1, −3, 17]

We call the rows R 1, R 2 and R 3.

2 Get 0s in column 1

→ [1, 2, 1, 5; 0, 1, 2, −3; 0, −3, −5, 7]

Careful!

A row move works on the whole row, the number after the bar too.

That number is the right side of the equation.

Careful: the whole row, after the bar too

3 Get 0s in column 2

→ [1, 0, −3, 11; 0, 1, 2, −3; 0, 0, 1, −2]

4 Get 0s in column 3

→ [1, 0, 0, 5; 0, 1, 0, 1; 0, 0, 1, −2]

5 Read the answer

Each row is an equation again.

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

So x = 5.

x = 5

Row 2: y = 1.

y = 1

Row 3: z = −2.

z = −2

We write it as (5, 1, −2).

(x, y, z) = (5, 1, −2)

6 Check

Let's check in every equation.

We put x = 5, y = 1, z = −2 into ①.

① 5 + 2(1) + (−2) = 5 + 2 − 2 = 5

We get 5, the same as the right side.

Now ②.

② 5 + 3(1) + 3(−2) = 5 + 3 − 6 = 2

We get 2, the same as the right side.

Now ③.

③ 2(5) + 1 − 3(−2) = 10 + 1 + 6 = 17

We get 17, 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 0s in column 2.

Then: get 0s in column 3.

Then: read the answer.

Last: check.

Answer