Question

The cost of $2$ books, $6$ notebooks and $3$ pens is $Rs.40$. The cost of $3$ books, $4$ notebooks and $2$ pens is $Rs.35$. While the cost of $5$ books, $7$ notebooks & $4$ pens is $Rs.61$. Using these informations and matrix method. Find the cost of two book, three notebooks and two separately.

Answer 1

Let book, notebook, pen be x, y, z respectively.

$2x+6y+3z=40$

$3x+4y+2z=35$

$5x+7y+4z=61$

$\left[\begin{array}{ccc}2& 6& 3\\ 3& 4& 2\\ 5& 7& 4\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}40\\ 35\\ 61\end{array}\right]$

Apply $R_1\to R_1-R_2$

$\left[\begin{array}{ccc}-1& 2& 1\\ 3& 4& 2\\ 5& 7& 4\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}5\\ 35\\ 61\end{array}\right]$

Now $R_1\to (-R_1)$

$\left[\begin{array}{ccc}1& -2& -1\\ 3& 4& 2\\ 5& 7& 4\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-5\\ 35\\ 61\end{array}\right]$

$R_2\to R_2-3R_1$ & $R_3\to R_3-5R_1$

$\left[\begin{array}{ccc}1& -2& -1\\ 0& 10& 5\\ 0& 17& 9\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-5\\ 50\\ 86\end{array}\right]$

$R_2\to \frac{1}{10}{R}_2,\left[\begin{array}{ccc}1& -2& -1\\ 0& 1& -\frac{1}{2}\\ 0& 17& 9\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-5\\ 5\\ 86\end{array}\right]$

$R_3-17R_2$, $\left[\begin{array}{ccc}1& -2& -1\\ 0& 1& \frac{1}{2}\\ 0& 0& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-5\\ 5\\ 1\end{array}\right]$

Now $x-2y-z=5$, $y+\frac{z}{2}=5$, $\frac{z}{2}=1$

$\Rightarrow z=2,y=4,x=5$.