Solve 2x + 4y = 3, x - 2y = 1 by Cramer's rule

Step by step

The question

Question: solve 2x + 4y = 3 and x − 2y = 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 = 2 × (−2) − 1 × 4.

2 times (−2) is −4.

1 times 4 is 4.

−4 minus 4 is −8.

So D = −8.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 3 and 1.

Careful!

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

The y column stays.

So D x = 3 × (−2) − 1 × 4.

3 times (−2) is −6.

1 times 4 is 4.

−6 minus 4 is −10.

So D x = −10.

3 Find Dy

D y: now the y column is replaced by 3 and 1.

So D y = 2 × 1 − 1 × 3.

2 times 1 is 2.

1 times 3 is 3.

2 minus 3 is −1.

So D y = −1.

4 Divide

x is D x over D: −10 over (−8).

Made simpler: x = 5/4.

y is D y over D: −1 over (−8).

Made simpler: y = 1/8.

So x = 5/4 and y = 1/8, both at the same time.

We write it as (5/4, 1/8).

5 Check

Let's check in every equation.

We put x = 5/4, y = 1/8 into ①.

We get 3, 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.