Use Gauss elimination to solve 4x1 x2 x3 2 5x1 x2 2x3

Use Gauss elimination to solve: 4x_1 + x_2 - x_3 = -2 5x_1 + x_2 + 2x_3 = 4 6x_1 + x_2 + x_3 = 6 Employ partial pivoting and check your answers by substituting them into the original equations.

Solution

4x₁ + x₂ - x₃ = -25

x₁ + x₂ + 2x₃ = 4

6x₁ + x₂ + x₃ = 6

Rewrite the system in matrix form and solve it by Gaussian Elimination (Gauss-Jordan elimination)

  4 1 -1 -2

5 1 2 4

6 1 1 6

R₁ R₁ / 4

1 0.25 -0.25 -0.5

5 1 2 4

6 1 1 6

R₂ R₂ - 5 R₁ ;

R₃ R₃ - 6 R₁

1 0.25 -0.25 -0.5

0   -0.25 3.25 6.5

0 -0.5 2.5 9

R₂ R₂ / -0.25

1 0.25 -0.25 -0.5

0 1 -13   -26

0 -0.5 2.5 9

R₁ R₁ - 0.25 R₂;

R₃ 0.5 R₂ + R₃

1 0 3 6

0 1 -13 -26

0 0 -4 -4

R₃ / -4 R₃

1 0 3 6

0 1 -13 -26

0 0 1 1

R₁ R₁ - 3 R₃;

R₂ 13R₃ + R₂

1 0 0 3

0 1 0 -13

0 0 1 1

x₁ = 3x₂ = -13x₃ = 1

Make a check:

4·3 + (-13) - 1 = 12 - 13 - 1 = -2
5·3 + (-13) + 2·1 = 15 - 13 + 2 = 4
6·3 + (-13) + 1 = 18 - 13 + 1 = 6

Check completed successfully.

Answer:

x₁ = 3

x₂ = -13

x₃ = 1