Solve 2x + 4y = -2, 4x - y = 5 with a matrix Step by Step

Step by step

The question

Question: solve 2x + 4y = −2 and 4x − y = 5 with a matrix.

2x + 4y = −2①

We want the x and y that make both true.

4x − y = 5②

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, 4, −2; 4, −1, 5]

We call the rows R 1 and R 2.

2 Get a 1 in row 1

→ [1, 2, −1; 4, −1, 5]

3 Get 0s in column 1

→ [1, 2, −1; 0, −9, 9]

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, −1; 0, 1, −1]

5 Get 0s in column 2

→ [1, 0, 1; 0, 1, −1]

6 Read the answer

Each row is an equation again.

Row 1 is 1x + 0y = 1.

So x = 1.

x = 1

Row 2: y = −1.

y = −1

We write it as (1, −1).

(x, y) = (1, −1)

7 Check

Let's check in every equation.

We put x = 1, y = −1 into ①.

① 2(1) + 4(−1) = 2 − 4 = −2

We get −2, the same as the right side.

Now ②.

② 4(1) − (−1) = 4 + 1 = 5

We get 5, 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