Solve x + y = 7, x - y = 1 by Cramer's rule

Step by step

The question

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

1 times (−1) is −1.

1 times 1 is 1.

−1 minus 1 is −2.

So D = −2.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 7 and 1.

Careful!

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

The y column stays.

So D x = 7 × (−1) − 1 × 1.

7 times (−1) is −7.

1 times 1 is 1.

−7 minus 1 is −8.

So D x = −8.

3 Find Dy

D y: now the y column is replaced by 7 and 1.

So D y = 1 × 1 − 1 × 7.

1 times 1 is 1.

1 times 7 is 7.

1 minus 7 is −6.

So D y = −6.

4 Divide

x is D x over D: −8 over (−2).

−8 divided by (−2) is 4.

So x = 4.

y is D y over D: −6 over (−2).

−6 divided by (−2) is 3.

So y = 3.

So x = 4 and y = 3, both at the same time.

We write it as (4, 3).

5 Check

Let's check in every equation.

We put x = 4, y = 3 into ①.

We get 7, the same as the right side.

Now ②.

We get 1, 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.