Step by step
The question
Question: solve x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 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 × (−1) × (−1), 1 × 1 × 1, 1 × 2 × 2.
1, 1, 4.
Together: 6.
Up: 1 × (−1) × 1, 2 × 1 × 1, −1 × 2 × 1.
−1, 2, −2.
Together: −1.
Down minus up: 6 minus (−1) is 7.
So D = 7.
2 Find Dx
D x: the x column is replaced by the right sides, 6, 3, 2.
Down: 6 × (−1) × (−1), 1 × 1 × 2, 1 × 3 × 2.
6, 2, 6.
Together: 14.
Up: 2 × (−1) × 1, 2 × 1 × 6, −1 × 3 × 1.
−2, 12, −3.
Together: 7.
Down minus up: 14 minus 7 is 7.
So D x = 7.
3 Find Dy
D y: the y column is replaced by the right sides, 6, 3, 2.
Down: 1 × 3 × (−1), 6 × 1 × 1, 1 × 2 × 2.
−3, 6, 4.
Together: 7.
Up: 1 × 3 × 1, 2 × 1 × 1, −1 × 2 × 6.
3, 2, −12.
Together: −7.
Down minus up: 7 minus (−7) is 14.
So D y = 14.
4 Find Dz
D z: the z column is replaced by the right sides, 6, 3, 2.
Careful!
For D z we replace the z column, the third one.
The other two stay.
Down: 1 × (−1) × 2, 1 × 3 × 1, 6 × 2 × 2.
−2, 3, 24.
Together: 25.
Up: 1 × (−1) × 6, 2 × 3 × 1, 2 × 2 × 1.
−6, 6, 4.
Together: 4.
Down minus up: 25 minus 4 is 21.
So D z = 21.
5 Divide
x is D x over D: 7/7 is 1.
y is D y over D: 14/7 is 2.
z is D z over D: 21/7 is 3.
So the answer is (1, 2, 3).
6 Check
Let's check in every equation.
We put x = 1, y = 2, z = 3 into ①.
We get 6, the same as the right side.
Now ②.
We get 3, the same as the right side.
Now ③.
We get 2, 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.