Solve x + 3y = 5, 2x - y = 10 by Cramer's rule

Step by step

The question

Question: solve x + 3y = 5 and 2x − y = 10 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 × 3.

1 times (−1) is −1.

2 times 3 is 6.

−1 minus 6 is −7.

So D = −7.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 5 and 10.

Careful!

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

The y column stays.

So D x = 5 × (−1) − 10 × 3.

5 times (−1) is −5.

10 times 3 is 30.

−5 minus 30 is −35.

So D x = −35.

3 Find Dy

D y: now the y column is replaced by 5 and 10.

So D y = 1 × 10 − 2 × 5.

1 times 10 is 10.

2 times 5 is 10.

10 minus 10 is 0.

So D y = 0.

4 Divide

x is D x over D: −35 over (−7).

−35 divided by (−7) is 5.

So x = 5.

y is D y over D: 0 over (−7).

0 divided by (−7) is 0.

So y = 0.

So x = 5 and y = 0, both at the same time.

We write it as (5, 0).

5 Check

Let's check in every equation.

We put x = 5, y = 0 into ①.

We get 5, the same as the right side.

Now ②.

We get 10, 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.