Solve 3x + y = 0, x + 2y = 5 by Cramer's rule

Step by step

The question

Question: solve 3x + y = 0 and x + 2y = 5 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 = 3 × 2 − 1 × 1.

3 times 2 is 6.

1 times 1 is 1.

6 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: 0 and 5.

Careful!

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

The y column stays.

So D x = 0 × 2 − 5 × 1.

0 times 2 is 0.

5 times 1 is 5.

0 minus 5 is −5.

So D x = −5.

3 Find Dy

D y: now the y column is replaced by 0 and 5.

So D y = 3 × 5 − 1 × 0.

3 times 5 is 15.

1 times 0 is 0.

15 minus 0 is 15.

So D y = 15.

4 Divide

x is D x over D: −5/5.

−5 divided by 5 is −1.

So x = −1.

y is D y over D: 15/5.

15 divided by 5 is 3.

So y = 3.

So x = −1 and y = 3, both at the same time.

We write it as (−1, 3).

5 Check

Let's check in every equation.

We put x = −1, y = 3 into ①.

We get 0, the same as the right side.

Now ②.

We get 5, 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.