Solve x - y = 3, 4x + y = 2 with a matrix Step by Step

Step by step

The question

Question: solve x − y = 3 and 4x + y = 2 with a matrix.

x − y = 3①

We want the x and y that make both true.

4x + y = 2②

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.

= [1, −1, 3; 4, 1, 2]

We call the rows R 1 and R 2.

2 Get 0s in column 1

→ [1, −1, 3; 0, 5, −10]

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 a 1 in row 2

→ [1, −1, 3; 0, 1, −2]

4 Get 0s in column 2

→ [1, 0, 1; 0, 1, −2]

5 Read the answer

Each row is an equation again.

Row 1 is 1x + 0y = 1.

So x = 1.

x = 1

Row 2: y = −2.

y = −2

We write it as (1, −2).

(x, y) = (1, −2)

6 Check

Let's check in every equation.

We put x = 1, y = −2 into ①.

① 1 − (−2) = 1 + 2 = 3

We get 3, the same as the right side.

Now ②.

② 4(1) + (−2) = 4 − 2 = 2

We get 2, the same as the right side.

Both 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: read the answer.

Last: check.

Answer