Question

Use the method of Gaussian elimination to solve the following systems of linear equations: 

Thanks guys i don't know how to do it with the extra variable

Answer 1

(a)

The given system of equations are

  x₁ -   x₂ -  7x₃  +  7x₄ = 5

 -x₁ +  x₂ +  8x₃  -   5x₄ = -7

3x₁ - 2x₂ - 17x₃ + 13x₄ = 14

2x₁ -   x₂ - 11x₃ +   8x₄ = 7

We can write above equations are in matrix form as

R indicates Rows

C indicates Columns

R₂ = R₂+R₁

R₃ = R₃-3R₁

R₄ = R₄-2R₁

Interchange R₃ with R₂

R₄ = R₄-R₂

R₄ = R₄+R₃

R₄ = R₄/4

We can write above matrix is in equations form as

x₁ -   x₂ -  7x₃  +  7x₄ = 5      -----------------(1)

        x₂ +  4x₃  -  8x₄ = -1     -----------------(2)

                  x₃  + 2x₄ = -2     -----------------(3)

                            x₄ = -1     -----------------(4)

Substitute x₄ = -1 in equation (3)

x₃  + 2(-1) = -2

x₃  = -2 + 2

x₃  = 0

Substitute x₄ = -1 and x₃  = 0  in equation (2)

x₂ +  4(0)  - 8 (-1) = -1

x₂ = -1 - 8

x₂ = -9

Substitute x₂ = -9 , x₄ = -1 and x₃  = 0  in equation (1)

x₁ - (-9) -  7(0) +  7(-1) = 5

x₁ = 5 - 9 + 7

x₁ = 3

Solution :

x₁ = 3

x₂ = -9

x₃  = 0

x₄ = -1

Answer 2

(b)

The given system of equations are

  x₁ -   x₂ -  7x₃  +  7x₄ = 5

 -x₁ +  x₂ +  8x₃  -   5x₄ = -7

3x₁ - 2x₂ - 17x₃ + 13x₄ = 14

0 x₁ - 0 x₂ - 0  x₃ + 0  x₄ = 0

We can write above equations are in matrix form as

R indicates Rows

C indicates Columns

R₂ = R₂+R₁

R₃ = R₃-3R₁

Interchange R₃ with R₂

We can write above matrix is in equations form as

x₁ -   x₂ -  7x₃  +  7x₄ = 5           ----------------------------(1)

        x₂ +  4x₃  -  8x₄ = -1          ---------------------------(2)

                  x₃  + 2x₄ = -2          ---------------------------(3)

Let x₄ = k

Substitute x₄ = k in equation (3)

x₃  + 2(k) = -2

x₃  = -2 - 2k

Substitute x₄ = k and x₃  = -2 - 2k  in equation (2)

x₂ +  4(-2 - 2k)  - 8(k) = -1

x₂ = -1 + 8k + 8k + 8

x₂ = 7 + 16k

Substitute x₂ = 7+16k , x₄ = k and x₃  = -2 + 2k in equation (1)

x₁ - (7+16k) -  7(-2 + 2k) +  7(k) = 5

x₁ = 5 + 7 +16k + 14 - 7k + 14k

x₁ = 26 + 23k

Solution :

x₁ = 26+23k

x₂ = 7+16k

x₃  = -2 - 2k

x₄ = k