Solve 2x - y = 5, x + 2y = 10 by Cramer's rule

Step by step

The question

Question: solve 2x − y = 5 and x + 2y = 10 by Cramer's rule.

We call them ① and ②.

We want the x and y that make both true.

Cramer's rule uses three determinants: D, D x and D y.

Then x is D x over D, and y is D y over D.

1 Find D

D: the numbers in front of x and y, in the same places as in the equations.

So D = 2 × 2 − 1 × (−1).

2 times 2 is 4.

1 times (−1) is −1.

Taking away a negative is adding: 4 + 1.

4 minus (−1) is 5.

So D = 5.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 5 and 10.

Careful!

For D x, we replace the x column: the first one.

The y column stays.

So D x = 5 × 2 − 10 × (−1).

5 times 2 is 10.

10 times (−1) is −10.

Taking away a negative is adding: 10 + 10.

10 minus (−10) is 20.

So D x = 20.

3 Find Dy

D y: now the y column is replaced by 5 and 10.

So D y = 2 × 10 − 1 × 5.

2 times 10 is 20.

1 times 5 is 5.

20 minus 5 is 15.

So D y = 15.

4 Divide

x is D x over D: 20/5.

20 divided by 5 is 4.

So x = 4.

y is D y over D: 15/5.

15 divided by 5 is 3.

So y = 3.

So x = 4 and y = 3, both at the same time.

We write it as (4, 3).

5 Check

Let's check in every equation.

We put x = 4, y = 3 into ①.

We get 5, the same as the right side.

Now ②.

We get 10, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: find D.

Then: find Dx.

Then: find Dy.

Then: divide.

Last: check.