Solve 3x + y + z = 11, x + y - z = 7, 2x + y + 3z = 6 by Cramer's rule Step by Step

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The question

Question: solve 3x + y + z = 11, x + y − z = 7, 2x + y + 3z = 6 by Cramer's rule.

3x + y + z = 11①

x + y − z = 7②

Three equations, three letters: x, y and z.

2x + y + 3z = 6③

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.

x = D_xD, y = D_yD, z = D_zD

1 Find D

D: the numbers in front of x, y and z, in their places.

D = |3, 1, 1; 1, 1, −1; 2, 1, 3|3, 1; 1, 1; 2, 1

We write the first two columns again, on the right.

Then every diagonal has three numbers.

Down: 3 × 1 × 3, 1 × (−1) × 2, 1 × 1 × 1.

down = 3 × 1 × 3 + 1 × −1 × 2 + 1 × 1 × 1

9, −2, 1.

Together: 8.

= 9 + (−2) + 1 = 8

Up: 2 × 1 × 1, 1 × (−1) × 3, 3 × 1 × 1.

up = 2 × 1 × 1 + 1 × −1 × 3 + 3 × 1 × 1

2, −3, 3.

Together: 2.

= 2 + (−3) + 3 = 2

Down minus up: 8 minus 2 is 6.

So D = 6.

D = 8 − 2 = 6

2 Find Dx

D x: the x column is replaced by the right sides, 11, 7, 6.

D_x = |11, 1, 1; 7, 1, −1; 6, 1, 3|11, 1; 7, 1; 6, 1

Down: 11 × 1 × 3, 1 × (−1) × 6, 1 × 7 × 1.

down = 11 × 1 × 3 + 1 × −1 × 6 + 1 × 7 × 1

33, −6, 7.

Together: 34.

= 33 + (−6) + 7 = 34

Up: 6 × 1 × 1, 1 × (−1) × 11, 3 × 7 × 1.

up = 6 × 1 × 1 + 1 × −1 × 11 + 3 × 7 × 1

6, −11, 21.

Together: 16.

= 6 + (−11) + 21 = 16

Down minus up: 34 minus 16 is 18.

So D x = 18.

D_x = 34 − 16 = 18

3 Find Dy

D y: the y column is replaced by the right sides, 11, 7, 6.

D_y = |3, 11, 1; 1, 7, −1; 2, 6, 3|3, 11; 1, 7; 2, 6

Down: 3 × 7 × 3, 11 × (−1) × 2, 1 × 1 × 6.

down = 3 × 7 × 3 + 11 × −1 × 2 + 1 × 1 × 6

63, −22, 6.

Together: 47.

= 63 + (−22) + 6 = 47

Up: 2 × 7 × 1, 6 × (−1) × 3, 3 × 1 × 11.

up = 2 × 7 × 1 + 6 × −1 × 3 + 3 × 1 × 11

14, −18, 33.

Together: 29.

= 14 + (−18) + 33 = 29

Down minus up: 47 minus 29 is 18.

So D y = 18.

D_y = 47 − 29 = 18

4 Find Dz

D z: the z column is replaced by the right sides, 11, 7, 6.

D_z = |3, 1, 11; 1, 1, 7; 2, 1, 6|3, 1; 1, 1; 2, 1

Careful!

For D z we replace the z column, the third one.

The other two stay.

Careful: Dz: the third column is replaced

Down: 3 × 1 × 6, 1 × 7 × 2, 11 × 1 × 1.

down = 3 × 1 × 6 + 1 × 7 × 2 + 11 × 1 × 1

18, 14, 11.

Together: 43.

= 18 + 14 + 11 = 43

Up: 2 × 1 × 11, 1 × 7 × 3, 6 × 1 × 1.

up = 2 × 1 × 11 + 1 × 7 × 3 + 6 × 1 × 1

22, 21, 6.

Together: 49.

= 22 + 21 + 6 = 49

Down minus up: 43 minus 49 is −6.

So D z = −6.

D_z = 43 − 49 = −6

5 Divide

x is D x over D: 18/6 is 3.

x = D_xD = 186 = 3

y is D y over D: 18/6 is 3.

y = D_yD = 186 = 3

z is D z over D: −6/6 is −1.

z = D_zD = −66 = −1

So the answer is (3, 3, −1).

(x, y, z) = (3, 3, −1)

6 Check

Let's check in every equation.

We put x = 3, y = 3, z = −1 into ①.

① 3(3) + 3 + (−1) = 9 + 3 − 1 = 11

We get 11, the same as the right side.

Now ②.

② 3 + 3 − (−1) = 3 + 3 + 1 = 7

We get 7, the same as the right side.

Now ③.

③ 2(3) + 3 + 3(−1) = 6 + 3 − 3 = 6

We get 6, 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.

Answer