Question

A manufacturer produces two types of products X and Y. Each product is first processed in a machine M and then sent to another machine N for finishing. Each unit of X requires 20 minutes time on M and 10 minutes time on N. Each unit of Y requires 10 minutes on M and 20 minutes time in N. The total time available on each machine is 600 minutes. Calculate the number of units of two types of produced by constructing matrix equation of the form AX=B and then solving by the matrix inversion method.​

Answer 1

Given : A manufacturer produces two types of products X and Y.

Each unit of X requires 20 minutes time on M and 10 minutes time on N.    Each unit of Y requires 10 minutes on M and 20 minutes time in N. \The total time available on each machine is 600 minutes.

To Find :  the number of units of two types of produced by constructing matrix equation of the form AX=B and then solving by the matrix inversion method.​

Solution:

20X   + 10Y  =  600

10X   + 20Y  =  600

$\left[\begin{array}{ccc}20&10\\10&20\end{array}\right] \left[\begin{array}{ccc}X\\Y\end{array}\right] =\left[\begin{array}{ccc}600\\600\end{array}\right]$

20X   + 10Y  =  600

=> 2X + Y = 60

10X   + 20Y  =  600

   X + Y = 60

$\left[\begin{array}{ccc}2&1\\1&2\end{array}\right] \left[\begin{array}{ccc}X\\Y\end{array}\right] =\left[\begin{array}{ccc}60\\60\end{array}\right]$

AX = B

X = A⁻¹B

$\left[\begin{array}{ccc}X\\Y\end{array}\right] =\left[\begin{array}{ccc}2&1\\1&2\end{array}\right]^{-1} \left[\begin{array}{ccc}60\\60\end{array}\right]$

A⁻¹ = AdjA/|A|

|A| = 2 * 2 - 1*1  = 3

$\left[\begin{array}{ccc}2&1\\1&2\end{array}\right]^{-1} = \dfrac{1}{3} \left[\begin{array}{ccc}2&-1\\-1&2\end{array}\right]$

$\left[\begin{array}{ccc}X\\Y\end{array}\right] = \dfrac{1}{3} \left[\begin{array}{ccc}2&-1\\-1&2\end{array}\right]\left[\begin{array}{ccc}60\\60\end{array}\right]$

$\left[\begin{array}{ccc}X\\Y\end{array}\right] = \dfrac{1}{3} \left[\begin{array}{ccc}60\\60\end{array}\right]$

$\left[\begin{array}{ccc}X\\Y\end{array}\right] = \left[\begin{array}{ccc}20\\20\end{array}\right]$

Units of each type = 20

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