Solve 2x + 3y = 13, x + y = 5 by Cramer's rule

Step by step

The question

Question: solve 2x + 3y = 13 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 = 2 × 1 − 1 × 3.

2 times 1 is 2.

1 times 3 is 3.

2 minus 3 is −1.

So D = −1.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 13 and 5.

Careful!

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

The y column stays.

So D x = 13 × 1 − 5 × 3.

13 times 1 is 13.

5 times 3 is 15.

13 minus 15 is −2.

So D x = −2.

3 Find Dy

D y: now the y column is replaced by 13 and 5.

So D y = 2 × 5 − 1 × 13.

2 times 5 is 10.

1 times 13 is 13.

10 minus 13 is −3.

So D y = −3.

4 Divide

x is D x over D: −2 over (−1).

−2 divided by (−1) is 2.

So x = 2.

y is D y over D: −3 over (−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 13, 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.