Question

The cost of $4$ dozen pencils, $3$ dozen pens and $2$ dozen erasers is Rs. $60$. The cost of $2$ dozen pencils, $4$ dozen pens and $6$ dozen erasers is Rs. $90$ whereas the cost of $6$ dozen pencils, $2$ dozen pens and $3$ dozen erasers is Rs. $70$. Find the cost of each items per dozen by using matrices

Answer 1

Let the cost of one dozen pencils, one dozen pends and one dozen erasers be $Rs.x,Rs.y$ and $Rs.z$ respectively.

According to the questions,

$4x+3y+2z=60$

$2x+4y+6z=90$

$6x+2y+3z=70$

The above equations can be expressed in the matrix form as

$\left[\begin{array}{ccc}4& 3& 2\\ 2& 4& 6\\ 6& 2& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}60\\ 90\\ 70\end{array}\right]$

Using $R_2\to R_2-\frac{1}{2}{R}_1$ and $R_3\to R_3-\frac{3}{2}{R}_1$

$\left[\begin{array}{ccc}4& 3& 2\\ 0& \frac{5}{2}& 5\\ 0& -\frac{5}{2}& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}60\\ 60\\ -20\end{array}\right]$

Using $R_3\to R_3+R_2$, we get

$\left[\begin{array}{ccc}4& 3& 2\\ 0& \frac{5}{2}& 5\\ 0& 0& 5\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}60\\ 60\\ 40\end{array}\right]$
Now the equations can be rewrite in their original form as

$5z=40....\left(i\right)$

$\Rightarrow z=8$

$\frac{5}{2}y+5z=60....\left(ii\right)$

$4x+3y+2z=60....\left(iii\right)$

Putting the value of $z$ in equation (ii),

$\frac{5}{2}y+5\left(8\right)=60$

$y=\frac{(60-40)}{5}\times 2=8$

$\Rightarrow y=8$

Putting the values of $y$ and $z$ in equations (iii), we get

$4x+3\left(8\right)+2\left(8\right)=60$

$4x=60-24-16$

$x=\frac{20}{4}=5$

$\therefore x=5$

$\therefore$ The cost of each item per dozen are $Rs.5,Rs.8$ and $Rs.8$ respectively.