The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.

Home » The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.

The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.

Solution:

Suppose the price of 1kg potato, wheat and rice is $x$ , $y$ and $z$ respectively.
As per the question,
$4x + 3y + 2z = 60$
$x+ 2y + 3z = 45$
$6x + 2y + 3z = 70$
Now converting the equations into matrix form
$AX = B$
$\begin{array}{l} {\left[\begin{array}{lll} 4 & 3 & 2 \\ 1 & 2 & 3 \\ 6 & 2 & 3 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 60 \\ 45 \\ 70 \end{array}\right]} \\ 4 R_{2}-R_{1} \\ 2 R_{3}-3 R_{1} \\ {\left[\begin{array}{ccc} 4 & 3 & 2 \\ 0 & 5 & 10 \\ 0 & -5 & 0 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 60 \\ 120 \\ -40 \end{array}\right]} \end{array}$
Again converting into the equations we get
$\begin{array}{l} 4 x+3 y+2 z=60 \\ 5 y+10 z=120 \\ -5 y=-40 \\ Y=8 \\ 5 \times 8+10 z=120 \\ 10 z=120-40 \\ Z=8 \\ 4 x+3 \times 8+2 \times 8=60 \\ 4 x=60-24-16 \\ 4 x=20 \\ X=5 \end{array}$
$\therefore$ The cost of $1 \mathrm{~kg}$ potatoes, wheat and rice is Rs.5, Rs. 8 and Rs. 8 respectively.