Solve 3x - 2y = 4, 2x + y = 12 by Cramer's rule

Step by step

The question

Question: solve 3x − 2y = 4 and 2x + y = 12 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 − 2 × (−2).

3 times 1 is 3.

2 times (−2) is −4.

Taking away a negative is adding: 3 + 4.

3 minus (−4) is 7.

So D = 7.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 4 and 12.

Careful!

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

The y column stays.

So D x = 4 × 1 − 12 × (−2).

4 times 1 is 4.

12 times (−2) is −24.

Taking away a negative is adding: 4 + 24.

4 minus (−24) is 28.

So D x = 28.

3 Find Dy

D y: now the y column is replaced by 4 and 12.

So D y = 3 × 12 − 2 × 4.

3 times 12 is 36.

2 times 4 is 8.

36 minus 8 is 28.

So D y = 28.

4 Divide

x is D x over D: 28/7.

28 divided by 7 is 4.

So x = 4.

y is D y over D: 28/7.

28 divided by 7 is 4.

So y = 4.

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

We write it as (4, 4).

5 Check

Let's check in every equation.

We put x = 4, y = 4 into ①.

We get 4, the same as the right side.

Now ②.

We get 12, 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.