Solve x + y = 1, 4x + y = -2 with a matrix Step by Step

Step by step

The question

Question: solve x + y = 1 and 4x + y = −2 with a matrix.

x + y = 1①

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, 1; 4, 1, −2]

We call the rows R 1 and R 2.

2 Get 0s in column 1

→ [1, 1, 1; 0, −3, −6]

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, 1; 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 = 1

We get 1, 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