Question

Using gaussian elimination method find all solutions to the following system of linear equations.

2x₂ + 3x₃ + 4x₄ = 1

x₁ - 3x₂ + 4x₃ + 5x₄ = 2

-3x₁ + 10x₂ - 6x₃ -7x₄ = -4

Answer 1

$$\begin{bmatrix} 0 & 2 & 3 & 4 & \ \ \ 1 \\ 1 & -3 &4 & 5 & \ \ \ 2 \\ -3 & 10 & -6 & -7 & \ \ \ -4 \\ \end{bmatrix}$$

Swap rows 1 and 2

$$\begin{bmatrix} 1 & -3 & 4 & 5 & \ \ \ 2 \\ 0 & 2 & 3 & 4 & \ \ \ 1 \\ -3 & 10 & -6 & -7 & \ \ \ -4 \\ \end{bmatrix}$$

$R_3=R_3+3R_1$

$$\begin{bmatrix} 1 & -3 & 4 & 5 & \ \ \ 2 \\ 0 & 2 & 3 & 4 & \ \ \ 1 \\ 0 & 1 & 6 & 8 & \ \ \ 2 \\ \end{bmatrix}$$

$R_2=R_2/2$

$$\begin{bmatrix} 1 & -3 & 4 & 5 & \ \ \ 2 \\ 0 & 1 & 3/2 & 2 & \ \ \ 1/2 \\ 0 & 1 & 6 & 8 & \ \ \ 2 \\ \end{bmatrix}$$

$R_1=R_1+3R_2$

$$\begin{bmatrix} 1 & 0 & 17/2 & 11 & \ \ \ 7/ 2 \\ 0 & 1 & 3/2 & 2 & \ \ \ 1/2 \\ 0 & 1 & 6 & 8 & \ \ \ 2 \\ \end{bmatrix}$$

$R_3=R_3-R_2$

$$\begin{bmatrix} 1 & 0 & 17/2 & 11 & \ \ \ 7/ 2 \\ 0 & 1 & 3/2 & 2 & \ \ \ 1/2 \\ 0 & 0 & 9/2 & 6 & \ \ \ 3/2 \\ \end{bmatrix}$$

$R_3=2R_3/9$

$$\begin{bmatrix} 1 & 0 & 17/2 & 11 & \ \ \ 7/ 2 \\ 0 & 1 & 3/2 & 2 & \ \ \ 1/2 \\ 0 & 0 & 1 & 4/3 & \ \ \ 1/3 \\ \end{bmatrix}$$

$R_1=R_1-17R_3/2$

$$\begin{bmatrix} 1 & 0 & 0 & -1/3 & \ \ \ 2/ 3 \\ 0 & 1 & 3/2 & 2 & \ \ \ 1/2 \\ 0 & 0 & 1 & 4/3 & \ \ \ 1/3 \\ \end{bmatrix}$$

$R_2=R_2-3R_3/2$

$$\begin{bmatrix} 1 & 0 & 0 & -1/3 & \ \ \ 2/ 3 \\ 0 & 1 & 0 & 0 & \ \ \ 0 \\ 0 & 0 & 1 & 4/3 & \ \ \ 1/3 \\ \end{bmatrix}$$

$$\begin{matrix} x_1-\dfrac{1}{3}x_4=\dfrac{2}{3} \\ \\ x_2=0 \\ \\ x_3+\dfrac{4}{3}x_4=\dfrac{1}{3} \\ \\ \end{matrix}$$

Solution set:

$$\begin{matrix} x_1=\dfrac{2}{3} +\dfrac{1}{3}x_4 \\ \\ x_2=0 \\ \\ x_3=\dfrac{1}{3}-\dfrac{4}{3}x_4 \\ \\ x_4,\ free \\ \\ \end{matrix}$$