Solve 3x + 2y = 12, x + y = 5 by Cramer's rule

Step by step

The question

Question: solve 3x + 2y = 12 and x + y = 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 × 1 − 1 × 2.

3 times 1 is 3.

1 times 2 is 2.

3 minus 2 is 1.

So D = 1.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 12 and 5.

Careful!

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

The y column stays.

So D x = 12 × 1 − 5 × 2.

12 times 1 is 12.

5 times 2 is 10.

12 minus 10 is 2.

So D x = 2.

3 Find Dy

D y: now the y column is replaced by 12 and 5.

So D y = 3 × 5 − 1 × 12.

3 times 5 is 15.

1 times 12 is 12.

15 minus 12 is 3.

So D y = 3.

4 Divide

x is D x over D: 2/1.

2 divided by 1 is 2.

So x = 2.

y is D y over D: 3/1.

3 divided by 1 is 3.

So y = 3.

So x = 2 and y = 3, both at the same time.

We write it as (2, 3).

5 Check

Let's check in every equation.

We put x = 2, y = 3 into ①.

We get 12, 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.