Solve each of the following systems of equations using matrix method. $3x + 4y + 7z = 4$; $2x – y + 3z = - 3$; $x + 2y

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Solve each of the following systems of equations using matrix method.
$3x + 4y + 7z = 4$ ;
$2x - y + 3z = - 3$ ;
$x + 2y - 3z = 8$ .

Solve each of the following systems of equations using matrix method.
$3x + 4y + 7z = 4$ ;
$2x - y + 3z = - 3$ ;
$x + 2y - 3z = 8$ .

Solution:

We need to find: $-\mathrm{x}, \mathrm{y}, \mathrm{z}$
The given set of lines are : –
$\begin{array}{l} 3 x+4 y+7 z=4 \\ 2 x-y+3 z=-3 \\ x+2 y-3 z=8 \end{array}$
Now converting the following equations in matrix form,
$\mathrm{Ax}=\mathrm{B}$
$\left(\begin{array}{ccc} 3 & 4 & 7 \\ 2 & -1 & 3 \\ 1 & 2 & -3 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{c} 4 \\ -3 \\ 8 \end{array}\right)$
$\begin{array}{l} 3 \mathrm{R}_{2}-2 \mathrm{R}_{1} \\ 3 \mathrm{R}_{3}-\mathrm{R}_{1} \\ \left(\begin{array}{ccc} 3 & 4 & 7 \\ 3 & -11 & -5 \\ 0 & 2 & -16 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{c} 4 \\ -17 \\ 20 \end{array}\right) \\ 11 \mathrm{R}_{3}+2 \mathrm{R}_{2} \\ \left(\begin{array}{ccc} 3 & 4 & 7 \\ 3 & -11 & -5 \\ 0 & 0 & -186 \end{array}\right)\left(\begin{array}{c} x \\ y \\ z \end{array}\right)=\left(\begin{array}{c} 4 \\ -17 \\ 186 \end{array}\right) \end{array}$
Again on converting back into equations we obtain
$\begin{array}{l} 3 x+4 y+7 z=4 \\ -11 y-5 z=-17 \\ -186 z=186 \\ z=-1 \end{array}$
$-11 y+5=-17$
$\begin{array}{l} -11 y=-22 \\ y=2 \end{array}$
$3 x+4 \times 2+7 \times-1=4$
$\begin{array}{l} 3 x=4-8+7 \\ x=1 \end{array}$
$\therefore x=1, y=2, z=-1$