Answer:
[tex](x,y,z)=(1477, 1464, 1437)[/tex]
Step-by-step explanation:
Consider the selling of the units positive earning and the purchasing of the units negative earning.
Case-1:
- Mr. A purchases 4 units of Z and sells 3 units of X and 5 units of Y
- Mr.A earns Rs6000
So, the equation would be
[tex]3x + 5y - 4z = 6000[/tex]
Case-2:
- Mr. B purchases 3 units of Y and sells 2 units of X and 1 units of Z
- Mr B neither lose nor gain meaning he has made 0₹
hence,
[tex]2x - 3y + z = 0[/tex]
Case-3:
- Mr. C purchases 1 units of X and sells 4 units of Y and 6 units of Z
- Mr.C earns 13000₹
therefore,
[tex] - x + 4y + 6z = 13000[/tex]
Thus our system of equations is
[tex]\begin{cases}3x + 5y - 4z = 6000\\2x - 3y + z = 0\\ - x + 4y + 6z = 13000\end{cases}[/tex]
Solving the system of equations:
we will consider elimination method to solve the system of equations. To do so ,separate the equation in two parts which yields:
[tex]\begin{cases}3x + 5y - 4z = 6000\\2x - 3y + z = 0\end{cases}\\\begin{cases}2x - 3y + z = 0\\ - x + 4y + 6z = 13000\end{cases}[/tex]
Now solve the equation accordingly:
[tex]\implies\begin{cases}11x-7y=6000\\-13x+22y=13000\end{cases}[/tex]
Solving the equation for x and y yields:
[tex]\implies\begin{cases}x= \dfrac{223000}{151}\\\\y= \dfrac{221000}{151}\end{cases}[/tex]
plug in the value of x and y into 2x - 3y + z = 0 and simplify to get z. hence,
[tex]\implies z= \dfrac{217000}{151} [/tex]
Therefore,the prices of commodities X,Y,Z are respectively approximately 1477, 1464, 1437
