Solve 4x - y = 7, x + y = 8 by Cramer's rule

Step by step

The question

Question: solve 4x − y = 7 and x + 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 = 4 × 1 − 1 × (−1).

4 times 1 is 4.

1 times (−1) is −1.

Taking away a negative is adding: 4 + 1.

4 minus (−1) is 5.

So D = 5.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 7 and 8.

Careful!

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

The y column stays.

So D x = 7 × 1 − 8 × (−1).

7 times 1 is 7.

8 times (−1) is −8.

Taking away a negative is adding: 7 + 8.

7 minus (−8) is 15.

So D x = 15.

3 Find Dy

D y: now the y column is replaced by 7 and 8.

So D y = 4 × 8 − 1 × 7.

4 times 8 is 32.

1 times 7 is 7.

32 minus 7 is 25.

So D y = 25.

4 Divide

x is D x over D: 15/5.

15 divided by 5 is 3.

So x = 3.

y is D y over D: 25/5.

25 divided by 5 is 5.

So y = 5.

So x = 3 and y = 5, both at the same time.

We write it as (3, 5).

5 Check

Let's check in every equation.

We put x = 3, y = 5 into ①.

We get 7, 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.