Solve x + 2y = 9, 2x - y = 3 by Cramer's rule

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The question

Question: solve x + 2y = 9 and 2x − y = 3 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 = 1 × (−1) − 2 × 2.

1 times (−1) is −1.

2 times 2 is 4.

−1 minus 4 is −5.

So D = −5.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 9 and 3.

Careful!

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

The y column stays.

So D x = 9 × (−1) − 3 × 2.

9 times (−1) is −9.

3 times 2 is 6.

−9 minus 6 is −15.

So D x = −15.

3 Find Dy

D y: now the y column is replaced by 9 and 3.

So D y = 1 × 3 − 2 × 9.

1 times 3 is 3.

2 times 9 is 18.

3 minus 18 is −15.

So D y = −15.

4 Divide

x is D x over D: −15 over (−5).

−15 divided by (−5) is 3.

So x = 3.

y is D y over D: −15 over (−5).

−15 divided by (−5) is 3.

So y = 3.

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

We write it as (3, 3).

5 Check

Let's check in every equation.

We put x = 3, y = 3 into ①.

We get 9, the same as the right side.

Now ②.

We get 3, 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.