Question

The cost of $4kg$ potato,$3kg$ wheat and $2kg$ rice is $Rs.150$. The cost of $1kg$ potato, $2kg$ potato, $2kg$ wheat and $3kg$ Rice is $Rs.125$. The cost of $6kg$ potato, $2kg$ wheat and $3kg$ Rice is $Rs.175$. Find the cost of each item $perkg$, by using Cramer's Rule.

Answer 1

Let the cost of $1$ Kg potato, $1$ Kg wheat and $1$ Kg rice be $x,y,z$ respectively

Then, ATQ:

$4x+3y+2z=150$

$x+2y+3z=125$

$6x+2y+3z=175$

$D=\left|\begin{array}{ccc}4& 3& 2\\ 1& 2& 3\\ 6& 2& 3\\ & & \end{array}\right|$

$=4(6-6)-3(3-18)+2(2-12)$

$=25$

$D_1=\left|\begin{array}{ccc}150& 3& 2\\ 125& 2& 3\\ 175& 2& 3\\ & & \end{array}\right|$

$=150(6-6)-125(9-4)+175(9-4)$

$=250$

$D_2=\left|\begin{array}{ccc}4& 150& 2\\ 1& 125& 3\\ 6& 175& 3\\ & & \end{array}\right|$

$=4(375-525)-150(3-18)+2(175-750)$

$=500$

$D_3=\left|\begin{array}{ccc}4& 3& 150\\ 1& 2& 125\\ 6& 2& 175\\ & & \end{array}\right|$

$=4(350-250)-3(175-750)+150(2-12)$

$=625$

Now,

$x=\frac{D_1}{D}=\frac{250}{25}=10$

$y=\frac{D_2}{D}=\frac{500}{25}=20$

$z=\frac{D_3}{D}=\frac{625}{25}=25$