Solve 4x - y = 3, x + 4y = 22 with a matrix

Step by step

The question

Question: solve 4x − y = 3 and x + 4y = 22 with a matrix.

We want the x and y that make both 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 and y, then the number on the right.

The bar stands for the = signs.

This is called an augmented matrix.

We call the rows R 1 and R 2.

2 Get a 1 in row 1

3 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.

4 Get a 1 in row 2

5 Get 0s in column 2

6 Read the answer

Each row is an equation again.

Row 1 is 1x + 0y = 2.

So x = 2.

Row 2: y = 5.

We write it as (2, 5).

7 Check

Let's check in every equation.

We put x = 2, y = 5 into ①.

We get 3, the same as the right side.

Now ②.

We get 22, 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.