Question

Solve the system of equations 2x + y + 5z = 18 5x + 3y – 2z = 2 x – 6y + 2z = 1 using Gauss elimination method with partial pivoting. Show all the steps.

Answer 1

$$\begin{pmatrix} 2 & 1 & 5 \\ 5 & 3 & -2 \\ 1 & -6 & 2 \\ \end{pmatrix}\begin{pmatrix} x \\ y \\ z\\ \end{pmatrix}=\begin{pmatrix} 18 \\ 2 \\ 1 \\ \end{pmatrix}$$

Augmented matrix

$$\begin{pmatrix} 2 & 1 & 5 & & 18 \\ 5 & 3 & -2 & & 2 \\ 1 & -6 & 2 & & 1 \\ \end{pmatrix}$$

$R_1\leftrightarrow R_2$

$$\begin{pmatrix} 5 & 3 & -2 & & 2 \\ 2 & 1 &5 & & 18 \\ 1 & -6 & 2 & & 1 \\ \end{pmatrix}$$

$R_2=R_2-2R_1/5$

$$\begin{pmatrix} 5 & 3 & -2 & & 2 \\ 0 & -1/5 & 29/5 & & 86/5\\ 1 & -6 & 2 & & 1 \\ \end{pmatrix}$$

$R_3=R_3-R_1/5$

$$\begin{pmatrix} 5 & 3 & -2 & & 2 \\ 0 & -1/5 & 29/5 & & 86/5\\ 0 & -33/5 & 12/5 & & 3/5 \\ \end{pmatrix}$$

$R_3=R_3-33R_2$

$$\begin{pmatrix} 5 & 3 & -2 & & 2 \\ 0 & -1/5 & 29/5 & & 86/5\\ 0 & 0 & -189 & & -567 \\ \end{pmatrix}$$

Back Substitute

$$5x+3y-2z=2$$

$$-\dfrac{1}{5}y+\dfrac{29}{5}z=\dfrac{86}{5}$$

$$-189z=-567$$

$$5x+3y-2(3)=2$$

$$-y+29(3)=86$$

$$z=3$$

$$5x+3(1)-6=2$$

$$y=1$$

$$z=3$$

$$x=1$$

$$y=1$$

$$z=3$$

$(1,1,3)$