Step by step
The question
Question: solve x + 3z = 5, 2x − y + 3z = 5, x + 3y + 3z = 11 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) × 3, 0 × 3 × 1, 3 × 2 × 3.
−3, 0, 18.
Together: 15.
Up: 1 × (−1) × 3, 3 × 3 × 1, 3 × 2 × 0.
−3, 9, 0.
Together: 6.
Down minus up: 15 minus 6 is 9.
So D = 9.
2 Find Dx
D x: the x column is replaced by the right sides, 5, 5, 11.
Down: 5 × (−1) × 3, 0 × 3 × 11, 3 × 5 × 3.
−15, 0, 45.
Together: 30.
Up: 11 × (−1) × 3, 3 × 3 × 5, 3 × 5 × 0.
−33, 45, 0.
Together: 12.
Down minus up: 30 minus 12 is 18.
So D x = 18.
3 Find Dy
D y: the y column is replaced by the right sides, 5, 5, 11.
Down: 1 × 5 × 3, 5 × 3 × 1, 3 × 2 × 11.
15, 15, 66.
Together: 96.
Up: 1 × 5 × 3, 11 × 3 × 1, 3 × 2 × 5.
15, 33, 30.
Together: 78.
Down minus up: 96 minus 78 is 18.
So D y = 18.
4 Find Dz
D z: the z column is replaced by the right sides, 5, 5, 11.
Careful!
For D z we replace the z column, the third one.
The other two stay.
Down: 1 × (−1) × 11, 0 × 5 × 1, 5 × 2 × 3.
−11, 0, 30.
Together: 19.
Up: 1 × (−1) × 5, 3 × 5 × 1, 11 × 2 × 0.
−5, 15, 0.
Together: 10.
Down minus up: 19 minus 10 is 9.
So D z = 9.
5 Divide
x is D x over D: 18/9 is 2.
y is D y over D: 18/9 is 2.
z is D z over D: 9/9 is 1.
So the answer is (2, 2, 1).
6 Check
Let's check in every equation.
We put x = 2, y = 2, z = 1 into ①.
We get 5, the same as the right side.
Now ②.
We get 5, the same as the right side.
Now ③.
We get 11, 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.