Step by step
The question
Question: solve x + 2y + z = 2, 3x + y = 10, x + y = 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 × 0, 2 × 0 × 1, 1 × 3 × 1.
0, 0, 3.
Together: 3.
Up: 1 × 1 × 1, 1 × 0 × 1, 0 × 3 × 2.
1, 0, 0.
Together: 1.
Down minus up: 3 minus 1 is 2.
So D = 2.
2 Find Dx
D x: the x column is replaced by the right sides, 2, 10, 2.
Down: 2 × 1 × 0, 2 × 0 × 2, 1 × 10 × 1.
0, 0, 10.
Together: 10.
Up: 2 × 1 × 1, 1 × 0 × 2, 0 × 10 × 2.
2, 0, 0.
Together: 2.
Down minus up: 10 minus 2 is 8.
So D x = 8.
3 Find Dy
D y: the y column is replaced by the right sides, 2, 10, 2.
Down: 1 × 10 × 0, 2 × 0 × 1, 1 × 3 × 2.
0, 0, 6.
Together: 6.
Up: 1 × 10 × 1, 2 × 0 × 1, 0 × 3 × 2.
10, 0, 0.
Together: 10.
Down minus up: 6 minus 10 is −4.
So D y = −4.
4 Find Dz
D z: the z column is replaced by the right sides, 2, 10, 2.
Careful!
For D z we replace the z column, the third one.
The other two stay.
Down: 1 × 1 × 2, 2 × 10 × 1, 2 × 3 × 1.
2, 20, 6.
Together: 28.
Up: 1 × 1 × 2, 1 × 10 × 1, 2 × 3 × 2.
2, 10, 12.
Together: 24.
Down minus up: 28 minus 24 is 4.
So D z = 4.
5 Divide
x is D x over D: 8/2 is 4.
y is D y over D: −4/2 is −2.
z is D z over D: 4/2 is 2.
So the answer is (4, −2, 2).
6 Check
Let's check in every equation.
We put x = 4, y = −2, z = 2 into ①.
We get 2, the same as the right side.
Now ②.
We get 10, 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.