Solve 2x - 5y = -1, 3x + 2y = 18 by Cramer's rule

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The question

Question: solve 2x − 5y = −1 and 3x + 2y = 18 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 × (−5).

2 times 2 is 4.

3 times (−5) is −15.

Taking away a negative is adding: 4 + 15.

4 minus (−15) is 19.

So D = 19.

2 Find Dx

D x: the same, but the x column is replaced by the numbers on the right: −1 and 18.

Careful!

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

The y column stays.

So D x = −1 × 2 − 18 × (−5).

−1 times 2 is −2.

18 times (−5) is −90.

Taking away a negative is adding: −2 + 90.

−2 minus (−90) is 88.

So D x = 88.

3 Find Dy

D y: now the y column is replaced by −1 and 18.

So D y = 2 × 18 − 3 × (−1).

2 times 18 is 36.

3 times (−1) is −3.

Taking away a negative is adding: 36 + 3.

36 minus (−3) is 39.

So D y = 39.

4 Divide

x is D x over D: 88/19.

Made simpler: x = 88/19.

y is D y over D: 39/19.

Made simpler: y = 39/19.

So x = 88/19 and y = 39/19, both at the same time.

We write it as (88/19, 39/19).

5 Check

Let's check in every equation.

We put x = 88/19, y = 39/19 into ①.

We get −1, the same as the right side.

Now ②.

We get 18, 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.