Solve each of the following systems of equations using matrix method. $2x + 8y + 5z = 5$; $x + y + z = - 2$; $x + 2y - z = 2$.

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Solve each of the following systems of equations using matrix method.
$2x + 8y + 5z = 5$ ;
$x + y + z = - 2$ ;
$x + 2y - z = 2$ .

Solve each of the following systems of equations using matrix method.
$2x + 8y + 5z = 5$ ;
$x + y + z = - 2$ ;
$x + 2y - z = 2$ .

Solution:

We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$
The given set of lines are:
$\begin{array}{l} 2 x+8 y+5 z=5 \\ x+y+z=-2 \\ x+2 y-z=2 \end{array}$
Now, converting the following equations in matrix form,
$\begin{array}{l} \mathrm{AX}=\mathrm{B} \\ \left(\begin{array}{ccc} 2 & 8 & 5 \\ 1 & 1 & 1 \\ 1 & 2 & -1 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{c} 5 \\ -2 \\ 2 \end{array}\right) \\ 2 \mathrm{R}_{2}-\mathrm{R}_{1} \\ 2 \mathrm{R}_{3}-\mathrm{R}_{1} \\ \left(\begin{array}{ccc} 2 & 8 & 5 \\ 0 & -6 & -3 \\ 0 & -4 & -7 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{c} 5 \\ -9 \\ -1 \end{array}\right) \\ 3 \mathrm{R}_{3}-2 \mathrm{R}_{2} \\ \left(\begin{array}{ccc} 2 & 8 & 5 \\ 0 & -6 & -3 \\ 0 & 0 & -15 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{c} 5 \\ -9 \\ 15 \end{array}\right) \end{array}$
Again on converting back into equations, we obtain
$\begin{array}{l} 2 x+8 y+5 z=5 \\ -6 y-3 z=-9 \end{array}$
$\begin{array}{l} -6 y-3 \times-1=-9 \\ -6 y=-9-3 \\ y=2 \end{array}$
$\begin{array}{l} -15 z=15 \\ z=-1 \end{array}$
$\begin{array}{l} 2 x+8 \times 2+5 \times-1=5 \\ 2 x=5-16+5 \\ x=-3 \end{array}$
$\therefore x=-3, y=2, z=-1$