Question

Solve the equations x + 2y + z = 7, 2x – y + 2z = 4, x + y – 2z = – 1 by using Cramer’s rule

Answer 1

x + 2y + z = 7

2x – y + 2z = 4

x + y – 2z = – 1

Here `Delta = |(1, 2, 1),(2, -1, 2),(1, 1,-2)|`

= 1(2 – 2) – 2(– 4 – 2) + 1(2 + 1)

= 1(0) – 2(– 6) + 1(3)

= 12 + 3

= 15 ≠ 0

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

`Delta_x = |(7, 2, 1),(4, -1, 2),(-1, 1, -2)|`

= 7(2 – 2) – 2(– 8 + 2) + 1(4 – 1)

= 7(0) – 2(–6) + 1(3)

= 12 + 3

= 15

`Delta_y = |(1, 7, 1),(2, 4, 2),(1, -1, -2)|`

= 1(– 8 + 2) – 7(– 4 – 2) + 1(– 2 – 4)

= 1(– 6) – 7(– 6) + 1(– 6)

= – 6 + 42 – 6

= 30

`Delta_z = |(1, 2, 7),(2, -1, 4),(1,1, -1)|`

= 1(1 – 4) – 2(– 2 – 4) + 7(2 + 1)

= 1(– 3) – 2(– 6) + 7(3)

= – 3 + 12 + 21

= 30

∴ By cramer’s rule

x =`Delta_x/Delta = 15/15` = 1

y =`Delta_y/Delta = 30/15` = 2

z =`Delta_z/Delta = 30/15` = 2

∴ The Solution is (x, y, z) = (1, 2, 2)