Question

Solve the following equations by using Cramer’s rule:

x + 4y + 3z = 2, 2x – 6y + 6z = – 3, 5x – 2y + 3z = – 5

Answer 1

The equations are

x + 4y + 3z = 2

2x – 6y + 6z =  – 3

5x – 2y + 3z =  – 5

Here `Delta = |(1, 4, 3),(2, -6, 6),(5, -2, 3)|`

= 1(– 18 + 12) – 4(6 – 30) + 3(– 4 + 30)

= 1(– 6) – 4(– 24) + 3(26)

= – 6 + 96 + 78

= 168 ≠ 0

∴ We can apply Cramer’s Rule and the system is consistent and it has unique solution.

`Delta_x = |(2, 4, 3),(-3, - 6, 6),(-5, -2, 3)|`

= 2(–18 + 12) – 4(– 9 + 30) + 3(6 – 30)

= 2(– 6) – 4(21) + 3(– 24)

= – 12 – 84 – 72

= – 168

`Delta_y = |(1, 2, 3),(2, -3, 6),(5, -5, 3)|`

= 1(– 9 + 30) – 2(6 – 30) + 3(– 10 + 15)

= 1(21) – 2(– 24) + 3(5)

= 21 + 48 + 15

= 84

`Delta_z = |(1, 4, 2),(2, -6, -3),(5, -2, -5)|`

= 1(30 – 6) – 4(– 10 +15) + 2(– 4 + 30)

= 1(24) – 4(5) + 2(26)

= 24 – 20 + 52

= 56

∴ By Cramer’s rule

x = `Delta_x/Delta = (- 168)/168` = – 1

y = `Delta_y/Delta = 84/168 = 1/2`

z = Delta_y/Delta = 56/168 = 1/3`

∴ (x, y, z) = `(-1, 1/2, 1/3)`