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#### Need solution for RD Sharma maths class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 19

\begin{aligned} &x=300, y=400, z=500 \end{aligned}

Given:

According to the question,

\begin{aligned} &x+y+z=1200 \\ &3 x+2 y+z=2200 \\ &4 x+y+3 z=3100 \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 that will be Adj A using Adj A calculate A-1.

Solution:

Let the award money given for discipline, politeness & punctuality be x, y, z respectively.

Since, the total cash award is 1200

\begin{aligned} &x+y+z=1200 \; \; \; \; \; \; \; ......(1)\\ \end{aligned}

Award money given by the school P is 2200

\begin{aligned} &3 x+2 y+z=2200 \; \; \; \; \; \; \; \; .......(2) \end{aligned}

Award money given by school Q is 3100

\begin{aligned} &4 x+y+3 z=3100 \; \; \; \; \; \; \; \; .......(3) \end{aligned}

The above system of equation can be written in matrix from AX=B

\begin{aligned} &{\left[\begin{array}{lll} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 1200 \\ 2200 \\ 3100 \end{array}\right]} \\ &\text { Where } A=\left[\begin{array}{lll} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3 \end{array}\right], X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \text { and } B=\left[\begin{array}{l} 1200 \\ 2200 \\ 3100 \end{array}\right] \end{aligned}

\begin{aligned} &\text { Now }|A|=\left|\begin{array}{lll} 1 & 1 & 1 \\ 3 & 2 & 1 \\ 4 & 1 & 3 \end{array}\right|=1(6-1)-1(9-4)+1(3-8)\\ &=5-5-5=-5 \neq 0\\ &\text { Let } c_{i j} \text { be the cofactors, } \end{aligned}

\begin{aligned} &c_{11}=(-1)^{1+1}\left|\begin{array}{ll} 2 & 1 \\ 1 & 3 \end{array}\right|=5, c_{12}=(-1)^{1+2}\left|\begin{array}{ll} 3 & 1 \\ 4 & 3 \end{array}\right|=-5 \\ &c_{13}=(-1)^{1+3}\left|\begin{array}{ll} 3 & 2 \\ 4 & 1 \end{array}\right|=-5, c_{21}=(-1)^{2+1}\left|\begin{array}{ll} 1 & 1 \\ 1 & 3 \end{array}\right|=-2 \end{aligned}

\begin{aligned} &c_{22}=(-1)^{2+2}\left|\begin{array}{ll} 1 & 1 \\ 4 & 3 \end{array}\right|=-1, c_{23}=(-1)^{2+13}\left|\begin{array}{ll} 1 & 1 \\ 4 & 1 \end{array}\right|=3 \\ &c_{31}=(-1)^{3+1}\left|\begin{array}{ll} 1 & 1 \\ 2 & 1 \end{array}\right|=-1, c_{32}=(-1)^{3+2}\left|\begin{array}{ll} 1 & 1 \\ 3 & 1 \end{array}\right|=2 \\ &c_{33}=(-1)^{3+3}\left|\begin{array}{ll} 1 & 1 \\ 3 & 2 \end{array}\right|=-1 \end{aligned}

\begin{aligned} &\operatorname{adj} A=\left[\begin{array}{ccc} 5 & -5 & -5 \\ -2 & -1 & 3 \\ -1 & 2 & -1 \end{array}\right]^{T}=\left[\begin{array}{ccc} 5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1 \end{array}\right] \\ &A^{-1}=\frac{1}{|A|} a d j A \end{aligned}

\begin{aligned} &=\frac{1}{-5}\left[\begin{array}{ccc} 5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1 \end{array}\right] \\ X &=A^{-1} B \end{aligned}

\begin{aligned} X &=\frac{1}{-5}\left[\begin{array}{ccc} 5 & -2 & -1 \\ -5 & -1 & 2 \\ -5 & 3 & -1 \end{array}\right]\left[\begin{array}{c} 1200 \\ 2200 \\ 3100 \end{array}\right] \\ X &=\frac{-1}{5}\left[\begin{array}{c} 6000-4400-3100 \\ -6000-2200+6200 \\ -6000+6600-3100 \end{array}\right] \end{aligned}

\begin{aligned} &{\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=-\frac{1}{5}\left[\begin{array}{l} -1500 \\ -2000 \\ -2500 \end{array}\right]} \\ &x=\frac{-1500}{-5}, y=\frac{-2000}{-5} \text { and } z=\frac{-2500}{-5} \\ &x=300, y=400, z=500 \end{aligned}

Hence, the award money for each value of tolerance kindness and leadership is Rs300Rs400 and Rs500 one more value which should be considered for award is honesty