Question

Solve the following equations by using Cramer’s rule:

2x + y – z = 3, x + y + z – 1, x – 2y – 3z = 4

Answer 1

The equations are

2x + y – z = 3

x + y + z = 1

x – 2y – 3z = 4

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

= 2(– 3 + 2) – 1(3 – 1) – 1(– 2 – 1)

= 2(–1) – 1(– 4) – 1(– 3)

= – 2 + 4 + 3

Δ = 5 ≠ 0

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

`Delta_x = |(3, 1, -1),(1, 1, 1),(4, -2, -3)|`

= 3(– 3 + 2) – 1(– 3 – 4) – 1(– 2 – 4)

= 3(– 1) – 1(– 7) – 1(6)

= – 3 + 7 + 6

= 10

`Delta_y = |(2, 3, -1),(1, 1, 1),(1, 4, -3)|`

= 2(– 3 – 4) – 3(– 3 – 1) – 1(4 – 1)

= 2(– 7) – 3(-4) – 1(3)

= – 14 + 12 – 3

= – 5

`Delta_-z = |(2, 1, 3),(1, 1, 1),(1, -2, 4)|`

= 2(4 + 2) – 1(4 – 1) + 3(– 2 – 1)

= 2(6) – 1(3) + 3(– 3)

= 12 – 3 – 9

= 0

∴ By Cramer’s rule

x = `Delta_x/Delta = 10/5` = 2

y = `Delta/Delta = (-5)/5` = – 1

z = `Delta_z/Delta = 0/5` = 0

∴ (x, y, z) = (2, – 1, 0)