Solve 2x + y = 15, x + 2y = 15 with a matrix Step by Step

Step by step

The question

Question: solve 2x + y = 15 and x + 2y = 15 with a matrix.

2x + y = 15①

We want the x and y that make both true.

x + 2y = 15②

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 and y, then the number on the right.

The bar stands for the = signs.

This is called an augmented matrix.

= [2, 1, 15; 1, 2, 15]

We call the rows R 1 and R 2.

2 Get a 1 in row 1

→ [1, 2, 15; 2, 1, 15]

3 Get 0s in column 1

→ [1, 2, 15; 0, −3, −15]

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

4 Get a 1 in row 2

→ [1, 2, 15; 0, 1, 5]

5 Get 0s in column 2

→ [1, 0, 5; 0, 1, 5]

6 Read the answer

Each row is an equation again.

Row 1 is 1x + 0y = 5.

So x = 5.

x = 5

Row 2: y = 5.

y = 5

We write it as (5, 5).

(x, y) = (5, 5)

7 Check

Let's check in every equation.

We put x = 5, y = 5 into ①.

① 2(5) + 5 = 10 + 5 = 15

We get 15, the same as the right side.

Now ②.

② 5 + 2(5) = 5 + 10 = 15

We get 15, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: write the augmented matrix.

Then: get a 1 in row 1.

Then: get 0s in column 1.

Then: get a 1 in row 2.

Then: get 0s in column 2.

Then: read the answer.

Last: check.

Answer