Step by step
The question
Question: solve x + 2y + 2z = 13, 3x + 3y + z = 10, x + 2y + z = 9 by Cramer's rule.
Three equations, three letters: x, y and z.
Cramer's rule, like with two equations: x is D x over D, y is D y over D, z is D z over D.
Four determinants, each by the diagonals.
1 Find D
D: the numbers in front of x, y and z, in their places.
We write the first two columns again, on the right.
Then every diagonal has three numbers.
Down: 1 × 3 × 1, 2 × 1 × 1, 2 × 3 × 2.
3, 2, 12.
Together: 17.
Up: 1 × 3 × 2, 2 × 1 × 1, 1 × 3 × 2.
6, 2, 6.
Together: 14.
Down minus up: 17 minus 14 is 3.
So D = 3.
2 Find Dx
D x: the x column is replaced by the right sides, 13, 10, 9.
Down: 13 × 3 × 1, 2 × 1 × 9, 2 × 10 × 2.
39, 18, 40.
Together: 97.
Up: 9 × 3 × 2, 2 × 1 × 13, 1 × 10 × 2.
54, 26, 20.
Together: 100.
Down minus up: 97 minus 100 is −3.
So D x = −3.
3 Find Dy
D y: the y column is replaced by the right sides, 13, 10, 9.
Down: 1 × 10 × 1, 13 × 1 × 1, 2 × 3 × 9.
10, 13, 54.
Together: 77.
Up: 1 × 10 × 2, 9 × 1 × 1, 1 × 3 × 13.
20, 9, 39.
Together: 68.
Down minus up: 77 minus 68 is 9.
So D y = 9.
4 Find Dz
D z: the z column is replaced by the right sides, 13, 10, 9.
Careful!
For D z we replace the z column, the third one.
The other two stay.
Down: 1 × 3 × 9, 2 × 10 × 1, 13 × 3 × 2.
27, 20, 78.
Together: 125.
Up: 1 × 3 × 13, 2 × 10 × 1, 9 × 3 × 2.
39, 20, 54.
Together: 113.
Down minus up: 125 minus 113 is 12.
So D z = 12.
5 Divide
x is D x over D: −3/3 is −1.
y is D y over D: 9/3 is 3.
z is D z over D: 12/3 is 4.
So the answer is (−1, 3, 4).
6 Check
Let's check in every equation.
We put x = −1, y = 3, z = 4 into ①.
We get 13, the same as the right side.
Now ②.
We get 10, the same as the right side.
Now ③.
We get 9, the same as the right side.
All of them work.
So the answer is right!
Let's look back.
First: find D.
Then: find Dx.
Then: find Dy.
Then: find Dz.
Then: divide.
Last: check.