Question

Solve the following linear equations by using Cramer’s rule.

x₁ - x₂ + x₃ = 2: x₁ + x₂ - x₃ = 0 : -x₁ - x₂ - x₃ = -6

Answer 1

∆ = `[(1,- 1,1),(1,1,-1)] [(x_1),(x_2),(x_3,)]`

= 1 (- 1 – 1) + 1 (- 1 – 1) + 1 (- 1 + 1)
= 1(- 2) + 1 (0)
∆ = – 2 – 2
∆ = – 2 – 2
∆ = – 4

`∆x_1` = `[(1,-1,2),(1,1,0),(-1,-1-,6)]`

= 2 (- 1 – 1) + 1 (0 + 6) + 1 (0 – 6)
= 2 (- 2) + 1(6) + 1 (- 6)
= – 4 + 6 – 6
∆ = – 4

`∆x_2` = `[(1,-1,2),(1,0,1),(-1,-1,-6)]`

= 1 (0 – 6) – 2 (- 1 – 1)+1(- 6 + 0)
= 1 (- 6) – 2 (- 2) + 1 (- 6)
= – 6 + 4 – 6
= – 12 + 4
∆x₃ = -8

`∆x_3` = `[(1,-1,2),(1,1,0),(-1,-1,-6)]`

1 (- 6 + 0) + 1 (- 6 + 0) + 2 (- 1 + 1)
= 1 (- 6) + 1 (- 6) + 2 (0)
= – 6 – 6
∆x₃ = – 12

X_{1 =} `"Δx"_1/Δ` ₌ `-4/-4` = 1

X₂ = `"∆x"_2/∆` = `- 8/-4` = 2

X_{3 =} `∆"x"_3/∆` _{= `-12/-4` = 3}

X₁ = 1, X₂ = 2, X₃ = 3

(x_1, x₂, x₃ ) = 1, 2, 3