#### Explain solution RD Sharma class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 20 maths

$x=R s 1125, y=R s 1125, z=R s 4750$

Given:

According to the question,

\begin{aligned} &x+x+y=7000 \\ &2 x+y=7000 \\ &26 x+17 y=110000 \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 amount deposited in each of the three accounts be Rs xRs y, Rs z respectively

Since the total amount deposited is Rs7000

\begin{aligned} &x+x+y=7000\\ &2 x+y=7000\; \; \; \; \; \; \; .......(1)\\ &\text { Total amount interest is } R s 550\\ &\frac{5}{100} x+\frac{8}{100} x+\frac{17}{200} y=550 \end{aligned}

\begin{aligned} &26 x+17 y=110000\; \; \; \; \; \; \; ......(2)\\ &\text { The above system equation can be written in matrix from } A X=B\\ &\left[\begin{array}{cc} 2 & 1 \\ 26 & 17 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 7000 \\ 11000 \end{array}\right]\\ &\text { Where } A=\left[\begin{array}{cc} 2 & 1 \\ 26 & 17 \end{array}\right] X=\left[\begin{array}{l} x \\ y \end{array}\right] \text { and } B=\left[\begin{array}{c} 7000 \\ 11000 \end{array}\right]\\ &\text { Now, }|A|=\left|\begin{array}{cc} 2 & 1 \\ 26 & 17 \end{array}\right|=34-26=8 \end{aligned}

$Let\; c_{i j}\; be\; the\; co\! f\! actors\; o\! f\; elements\; a_{i j}\; in\; A=\left[a_{i j}\right] \\ Then \\ c_{11}=(-1)^{1+1} 17=17, c_{12}=(-1)^{1+2} 26=-26 \\ c_{21}=(-1)^{2+1}(1)=-1, c_{22}=(-1)^{2+2}(2)=2$

\begin{aligned} &\operatorname{adj} A=\left[\begin{array}{cc} 17 & -26 \\ -1 & 2 \end{array}\right]^{T} \\ &=\left[\begin{array}{cc} 17 & -1 \\ -26 & 2 \end{array}\right] \\ &A^{-1}=\frac{1}{|A|} a d j A \\ &=\frac{1}{8}\left[\begin{array}{cc} 17 & -1 \\ -26 & 2 \end{array}\right] \end{aligned}

\begin{aligned} A^{-1} &=\frac{1}{|A|} a d j A \\ &=\frac{1}{8}\left[\begin{array}{cc} 17 & -1 \\ -26 & 2 \end{array}\right] \\ X=& A^{-1} B \end{aligned}

\begin{aligned} &{\left[\begin{array}{c} x \\ y \end{array}\right]=\frac{1}{8}\left[\begin{array}{cc} 17 & -1 \\ -26 & 2 \end{array}\right]\left[\begin{array}{c} 7000 \\ 110000 \end{array}\right]} \\ &{\left[\begin{array}{c} x \\ y \end{array}\right]=\frac{1}{8}\left[\begin{array}{c} 119000-110000 \\ -182000+220000 \end{array}\right]} \\ &{\left[\begin{array}{l} x \\ y \end{array}\right]=\frac{1}{8}\left[\begin{array}{c} 9000 \\ 38000 \end{array}\right]} \end{aligned}

\begin{aligned} &x=\frac{9000}{8} \text { and } y=\frac{38000}{8} \\ &x=1125 \text { and } y=4750 \end{aligned}

Hence the amount deposited in each of the three accounts is Rs1125, Rs1102, Rs4750