#### Provide solution for RD Sharma maths class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 14

\begin{aligned} &x=500, y=2000, z=3500 \end{aligned}

Given:

As per the information the following equation is

\begin{aligned} &x+y+z=6000 \\ &x+0 y+3 z=1100 \\ &x-2 y+z=0 \end{aligned}

Hint:

X=A-1B is used to solve this problem. And the determinant and co-factor of matrix A, take it’s transpose, and that will be Adj A using Adj A calculate A-1.

Solution:

Let the award money given for honesty regularity and hard work be x, y, z respectively

Since total cash award is  Rs6000

\begin{aligned} &x+y+z=6000 \: \: \: \: \: \: .....(1)\end{aligned}

Three times the award money for hard work and honesty is Rs11000

\begin{aligned} x+0 y+3 z=1100\: \: \: \: \: \: .......(2) \end{aligned}

Award money for honesty and hardwork is double the one given for regularity

\begin{aligned}&x+z=2y \\ &x-2 y+z=0\: \: \: \: \: \: .....(3) \end{aligned}

\begin{aligned} &\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 6000 \\ 11000 \\ 0 \end{array}\right]\\ &A X=B\\ &A=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & 3 \\ 1 & -2 & 1 \end{array}\right|=6 \neq 0 \text { Thus, } A \text { is non singular } \end{aligned}

$\begin{gathered} \operatorname{adj} A=\left[\begin{array}{ccc} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & 1 \end{array}\right] \\ A^{-1}=\frac{1}{|A|} a d j A=\frac{1}{6}\left[\begin{array}{ccc} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & 1 \end{array}\right] \end{gathered}$

\begin{aligned} &X=A^{-1} B \\ &=\frac{1}{6}\left[\begin{array}{ccc} 6 & -3 & 3 \\ 2 & 0 & -2 \\ -2 & 3 & 1 \end{array}\right]\left[\begin{array}{c} 6000 \\ 11000 \\ 0 \end{array}\right] \end{aligned}

\begin{aligned} &{\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 500 \\ 2000 \\ 3500 \end{array}\right]} \\ &\text { Hence, } x=500, y=2000, z=3500 \end{aligned}

Thus award money given for honesty, regularity and hardwork are Rs500, Rs2000 & Rs3500 respectively