Step by step
The question
Question: solve x + y + 2z = 11, x + 3z = 11, x + 2z = 7 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 × 0 × 2, 1 × 3 × 1, 2 × 1 × 0.
0, 3, 0.
Together: 3.
Up: 1 × 0 × 2, 0 × 3 × 1, 2 × 1 × 1.
0, 0, 2.
Together: 2.
Down minus up: 3 minus 2 is 1.
So D = 1.
2 Find Dx
D x: the x column is replaced by the right sides, 11, 11, 7.
Down: 11 × 0 × 2, 1 × 3 × 7, 2 × 11 × 0.
0, 21, 0.
Together: 21.
Up: 7 × 0 × 2, 0 × 3 × 11, 2 × 11 × 1.
0, 0, 22.
Together: 22.
Down minus up: 21 minus 22 is −1.
So D x = −1.
3 Find Dy
D y: the y column is replaced by the right sides, 11, 11, 7.
Down: 1 × 11 × 2, 11 × 3 × 1, 2 × 1 × 7.
22, 33, 14.
Together: 69.
Up: 1 × 11 × 2, 7 × 3 × 1, 2 × 1 × 11.
22, 21, 22.
Together: 65.
Down minus up: 69 minus 65 is 4.
So D y = 4.
4 Find Dz
D z: the z column is replaced by the right sides, 11, 11, 7.
Careful!
For D z we replace the z column, the third one.
The other two stay.
Down: 1 × 0 × 7, 1 × 11 × 1, 11 × 1 × 0.
0, 11, 0.
Together: 11.
Up: 1 × 0 × 11, 0 × 11 × 1, 7 × 1 × 1.
0, 0, 7.
Together: 7.
Down minus up: 11 minus 7 is 4.
So D z = 4.
5 Divide
x is D x over D: −1/1 is −1.
y is D y over D: 4/1 is 4.
z is D z over D: 4/1 is 4.
So the answer is (−1, 4, 4).
6 Check
Let's check in every equation.
We put x = −1, y = 4, z = 4 into ①.
We get 11, the same as the right side.
Now ②.
We get 11, the same as the right side.
Now ③.
We get 7, 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.