#### Please solve RD Sharma class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 21 maths textbook solution

$V\! ariety \; A: R s 5, V\! ariety\; B: R s 8, V\! ariety\; C: R s 8$

Given:

According to the question

$A+B+C=21,\;4 A+3 B+2 C=60,\;6 A+2 B+3 C=70$

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:

As there are 3 variables of pen A, B, C

Meenu purchased a pen of each variety which costs her Rs21

Therefore

$A+B+C=21$

Similarly

For Jeevan

$4 A+3 B+2 C=60$

For Shikha

$6 A+2 B+3 C=70$

\begin{aligned} &{\left[\begin{array}{lll} 1 & 1 & 1 \\ 4 & 3 & 2 \\ 6 & 2 & 3 \end{array}\right]\left[\begin{array}{l} A \\ B \\ C \end{array}\right]=\left[\begin{array}{l} 21 \\ 60 \\ 70 \end{array}\right]} \\ &\text { Where } P=\left[\begin{array}{lll} 1 & 1 & 1 \\ 4 & 3 & 2 \\ 6 & 2 & 3 \end{array}\right], Q=\left[\begin{array}{l} 21 \\ 60 \\ 70 \end{array}\right] \\ &|P|=1(9-4)-1(12-12)+1(8-18) \\ &=-5 \neq 0 \end{aligned}

\begin{aligned} &P^{-1} \text { exists }\\ &c_{11}=5 \quad c_{12}=0 \quad c_{13}=-10\\ &c_{21}=-1 \quad c_{22}=-3 \quad c_{23}=4\\ &c_{31}=-1 \quad c_{32}=2 \quad c_{33}=-1 \end{aligned}

\begin{aligned} &\operatorname{adjP}=\left[\begin{array}{ccc} 5 & 0 & -10 \\ -1 & -3 & 4 \\ -1 & 2 & -1 \end{array}\right]^{T} \\ &=\left[\begin{array}{ccc} 5 & -1 & -1 \\ 0 & -3 & 2 \\ -10 & 4 & -1 \end{array}\right] \\ &P^{-1}=\frac{1}{-5}\left[\begin{array}{ccc} 5 & -1 & -1 \\ 0 & -3 & 2 \\ -10 & 4 & -1 \end{array}\right] \end{aligned}

\begin{aligned} &X=P^{-1} Q \\ &=\frac{1}{-5}\left[\begin{array}{ccc} 5 & -1 & -1 \\ 0 & -3 & 2 \\ -10 & 4 & -1 \end{array}\right]\left[\begin{array}{l} 21 \\ 60 \\ 70 \end{array}\right] \\ &=\frac{1}{-5}\left[\begin{array}{c} 105-60-70 \\ 0-180+140 \\ -210+240-70 \end{array}\right] \end{aligned}

$\begin{gathered} =\frac{1}{-5}\left[\begin{array}{l} -25 \\ -40 \\ -40 \end{array}\right] \\ X=\left[\begin{array}{l} 5 \\ 8 \\ 8 \end{array}\right] \end{gathered}$

Therefore

Cost of A variety of pens= Rs5

Cost of B variety of pens= Rs8

Cost of C variety of pens= Rs8