Solve -x + y = 0, -3x + 4y = 2 with a matrix Step by Step

Step by step

The question

Question: solve −x + y = 0 and −3x + 4y = 2 with a matrix.

−x + y = 0①

We want the x and y that make both true.

−3x + 4y = 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, 0; −3, 4, 2]

We call the rows R 1 and R 2.

2 Get a 1 in row 1

→ [1, −1, 0; −3, 4, 2]

3 Get 0s in column 1

→ [1, −1, 0; 0, 1, 2]

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 0s in column 2

→ [1, 0, 2; 0, 1, 2]

5 Read the answer

Each row is an equation again.

Row 1 is 1x + 0y = 2.

So x = 2.

x = 2

Row 2: y = 2.

y = 2

We write it as (2, 2).

(x, y) = (2, 2)

6 Check

Let's check in every equation.

We put x = 2, y = 2 into ①.

① −2 + 2 = 0

We get 0, the same as the right side.

Now ②.

② −3(2) + 4(2) = −6 + 8 = 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 a 1 in row 1.

Then: get 0s in column 1.

Then: get 0s in column 2.

Then: read the answer.

Last: check.

Answer