Question

Solve the following equations by using Cramer’s rule:

x + y + z = 6, 2x + 3y – z = 5, 6x – 2y – 3z = – 7

Answer 1

The equations are

x + y + z = 6

2x + 3y – z = 5

6x – 2y – 3z = – 7

Here Δ = `|(1, 1, 1),(2, 3, -1),(6, -2, -3)|`

= 1(– 9 – 2) –1(– 6 + 6) + 1(– 4 – 18)

= 1(–11) – 1(0) +1(– 22)

= – 11 – 22

= – 33 ≠ 0

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

Now, `Delta_x = |(6, 1, 1),(5, 3, -1),(-7, -2, -3)|`

= 6(– 9 – 2) – 1(–15 – 7) + 1(–10 + 21)

= 6(– 11) – 1(– 22)+ 1 (11)

= – 66 + 22 + 11

= – 33

`Delta_y |(1, 6, 1),(2, 5, -1),(6, -7, -3)|`

= 1(– 15 – 7) – 6(– 6 + 6) + 1(– 14 – 30)

= 1(– 22) – 6(0) + 1(– 44)

= – 66

`Delta_z = |(1, 1, 6),(2, 3, 5),(6, -2, -7)|`

= 1(– 21 + 10) – 1(– 14 – 30) + 6(– 4 – 18)

=  (– 11) – 1(– 44) + 6(– 22)

= – 11 + 44 – 132

= – 99

∴ By Cramer’s rule

x = `Delta_x/Delta = (- 33)/(-33)` = 1

y = `Delta_y/Delta = (- 66)/(- 33)` = 2

z = `Delta_z/Delta = (- 99)/(- 33)` = 3

∴ (x, y, z) = (1, 2, 3)