Solve x + y = 4, 3x - y = 8 by Cramer's rule

Step by step

The question

Question: solve x + y = 4 and 3x − y = 8 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) − 3 × 1.

1 times (−1) is −1.

3 times 1 is 3.

−1 minus 3 is −4.

So D = −4.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 4 and 8.

Careful!

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

The y column stays.

So D x = 4 × (−1) − 8 × 1.

4 times (−1) is −4.

8 times 1 is 8.

−4 minus 8 is −12.

So D x = −12.

3 Find Dy

D y: now the y column is replaced by 4 and 8.

So D y = 1 × 8 − 3 × 4.

1 times 8 is 8.

3 times 4 is 12.

8 minus 12 is −4.

So D y = −4.

4 Divide

x is D x over D: −12 over (−4).

−12 divided by (−4) is 3.

So x = 3.

y is D y over D: −4 over (−4).

−4 divided by (−4) is 1.

So y = 1.

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

We write it as (3, 1).

5 Check

Let's check in every equation.

We put x = 3, y = 1 into ①.

We get 4, the same as the right side.

Now ②.

We get 8, 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.