Solve the system x1 4x3 3x4 31 x2 3x3 4x4 27 2x1 3x2

Solve the system {x_1 + 4x_3 + 3x_4 = -31 x_2 - 3x_3 - 4x_4 = 27 2x_1 - 3x_2 + 14x_3 + 18x_4 = -125 -x_2 + 3x_3 + 9x_4 = -42

Solution

let the given equations (1) ,(2), (3)and (4)

subtract the eq(3) from eq(1)multiplied by 2

2x₁ + +8x₃ + 6x₄ = -62

2x₁ -3x₂   +14x₃+18x₄ =-125

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   3x₂ -6x₃   -12x₄ = 63 _____________(5)

(+) -3x₂ +9x₃ +27x₄ = -126 _____________(4)x3

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3x₃ + 15x₄ = -63 ______________(6)

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3x₂ - 6x₃ -12x₄ = 63 ___________(5)

(-) 3x₂ -9x₃   -12x₄   = 81 ___________(2)x3

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3x₃ = -18

implies x₃ = -6

substituting x₃ in eq(6) we get x₄ = -3

substituting x₃ and x₄ in eq(5) we get x₂ = -3

substituting all these values in eq(1) we get x₁= 2

therefore x₁ = 2, x₂ = -3, x₃ = -6 and x₄ = -3.