Solve x - 3y = -7, 2x + y = 7 by Cramer's rule

Step by step

The question

Question: solve x − 3y = −7 and 2x + y = 7 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.

Taking away a negative is adding: 1 + 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: −7 and 7.

Careful!

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

The y column stays.

So D x = −7 × 1 − 7 × (−3).

−7 times 1 is −7.

7 times (−3) is −21.

Taking away a negative is adding: −7 + 21.

−7 minus (−21) is 14.

So D x = 14.

3 Find Dy

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

So D y = 1 × 7 − 2 × (−7).

1 times 7 is 7.

2 times (−7) is −14.

Taking away a negative is adding: 7 + 14.

7 minus (−14) is 21.

So D y = 21.

4 Divide

x is D x over D: 14/7.

14 divided by 7 is 2.

So x = 2.

y is D y over D: 21/7.

21 divided by 7 is 3.

So y = 3.

So x = 2 and y = 3, both at the same time.

We write it as (2, 3).

5 Check

Let's check in every equation.

We put x = 2, y = 3 into ①.

We get −7, the same as the right side.

Now ②.

We get 7, 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.