Question

A shopkeeper has $3$ varieties of pens $A,B$ and $C$. Meenu purchased $1$ pen of each variety for a total of Rs. $21$. Jeen purchased $4$ pens of $A$ variety, $3$ pens of $B$ variety and $2$ pens of $C$ variety for Rs. $60$. While Shikha purchased $6$ pens of $A$ variety, $2$ pens of $B$ variety and $3$ pens of $C$ variety for Rs. $70$. Using matrix method find the cost of each pen.

Answer 1

From the given information we can form a matrix as follows :

$\left[\begin{array}{ccc}1& 1& 1\\ 4& 3& 2\\ 6& 2& 3\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\end{array}\right]=\left[\begin{array}{c}21\\ 60\\ 70\end{array}\right]$

Applying $R_2\to R_2-4R_1,$ $R_3\to R_3-6R_1$

$\left[\begin{array}{ccc}1& 1& 1\\ 0& -1& -2\\ 0& -4& -3\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\end{array}\right]=\left[\begin{array}{c}21\\ -24\\ -56\end{array}\right]$

Applying $R_3\to R_3+(-4)R_2$

$\left[\begin{array}{ccc}1& 1& 1\\ 0& -1& -2\\ 0& 0& 5\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\end{array}\right]=\left[\begin{array}{c}21\\ -24\\ 40\end{array}\right]$

From above we get three equations,

$\Rightarrow$ $A+B+C=21$ ----- ( 1 )

$\Rightarrow$ $-B-2C=-24$ ----- ( 2 )

$\Rightarrow$ $5C=40$

$\therefore$ $C=8$ ------ ( 3 )

Substituting $C=8$ in equation ( 2 ), we get

$\Rightarrow$ $-B-2\left(8\right)=-24$

$\Rightarrow$ $-B=-24+16$

$\therefore$ $B=8$

Substituting value of $B$ and $C$ in equation ( 1 ) we get,

$\Rightarrow$ $A+8+8=21$

$\Rightarrow$ $A+16=21$

$\therefore$ $A=5$

$\therefore$ Cost of $A$ variety pen is $Rs.5,$ $B$ variety pen is $Rs.8$ and $C$ variety pen is $Rs.8$.