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

Step by step

The question

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

−2 times (−2) is 4.

3 times 1 is 3.

4 minus 3 is 1.

So D = 1.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: 1 and −4.

Careful!

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

The y column stays.

So D x = 1 × (−2) − (−4 × 1).

1 times (−2) is −2.

−4 times 1 is −4.

Taking away a negative is adding: −2 + 4.

−2 minus (−4) is 2.

So D x = 2.

3 Find Dy

D y: now the y column is replaced by 1 and −4.

So D y = −2 × (−4) − 3 × 1.

−2 times (−4) is 8.

3 times 1 is 3.

8 minus 3 is 5.

So D y = 5.

4 Divide

x is D x over D: 2/1.

2 divided by 1 is 2.

So x = 2.

y is D y over D: 5/1.

5 divided by 1 is 5.

So y = 5.

So x = 2 and y = 5, both at the same time.

We write it as (2, 5).

5 Check

Let's check in every equation.

We put x = 2, y = 5 into ①.

We get 1, the same as the right side.

Now ②.

We get −4, 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.