Question

Factories A and B sent rice to stores 1 and 2. A sent 14 loads and B sent 22. Store 1 used 20 loads and
store 2 used 16. It cost $200 to ship from A to 1, $350 from A to 2, $300 from B to 1, and $250 from B
to 2. $8600 was spent. How many loads went where ?

Answer 1

Factory A sent 14 loads of rice to Store 1

Factory A sent no loads of rice to Store 2

Factory B sent 6 loads of rice to Store 1

Factory B sent 16 loads of rice to Store 2

Let

    A1 = Loads sent from Factory A to Store 1

    A2 = Loads sent from Factory A to Store 2

    B1 = Loads sent from Factory B to Store 1

    B2 = Loads sent from Factory B to Store 2

Then, the equations describing the scenario are;

    $A1+A2=14$

    $B1+B2=22$

    $A1+B1=20$

    $A2+B2=16$

    $200A1+350A2+300B1+250B2=8600$

The simultaneous equations can be expressed in matrix form thus:

$\left[\begin{array}{ccccc}{A1}&{A2}&{B1}&{B2}&{}\\1&1&0&0&14\\0&0&1&1&22\\1&0&1&0&20\\0&1&0&1&16\\200&350&300&250&8600\end{array}\right]$

Reducing the matrix:

Step 1:

$R_3 \leftarrow R_3 - R_1\\R_5 \leftarrow R_5 - 200R_1$

$\left[\begin{array}{ccccc}{A1}&{A2}&{B1}&{B2}&{}\\1&1&0&0&14\\0&0&1&1&22\\0&-1&1&0&6\\0&1&0&1&16\\0&150&300&250&5800\end{array}\right]$

Step 2:

Switch $R_4$ and $R_2$

$\left[\begin{array}{ccccc}{A1}&{A2}&{B1}&{B2}&{}\\1&1&0&0&14\\0&1&0&1&16\\0&-1&1&0&6\\0&0&1&1&22\\0&150&300&250&5800\end{array}\right]$

Step 3:

$R_3\leftarrow R_3+R_2\\R_5 \leftarrow R_5-150R_2$

$\left[\begin{array}{ccccc}{A1}&{A2}&{B1}&{B2}&{}\\1&1&0&0&14\\0&1&0&1&16\\0&0&1&1&22\\0&0&1&1&22\\0&0&300&100&3400\end{array}\right]$

Step 4:

$R_4\leftarrow R_4+R_3\\R_5 \leftarrow R_5-300R_3$

$\left[\begin{array}{ccccc}{A1}&{A2}&{B1}&{B2}&{}\\1&1&0&0&14\\0&1&0&1&16\\0&0&1&1&22\\0&0&0&0&0\\0&0&0&-200&-3200\end{array}\right]$

Step 5:

Switch $R_4$ and $R_5$

$R_4\leftarrow R_4 \times \frac{1}{-200}\\R_1 \leftarrow R_1-R_2$

$\left[\begin{array}{ccccc}{A1}&{A2}&{B1}&{B2}&{}\\1&0&0&-1&-2\\0&1&0&1&16\\0&0&1&1&22\\0&0&0&1&16\\0&0&0&0&0\end{array}\right]$

Step 6:

$R_1 \leftarrow R_1 + R_4\\R_2\leftarrow R_2 - R_4\\R_3 \leftarrow R_3-R_4$

$\left[\begin{array}{ccccc}{A1}&{A2}&{B1}&{B2}&{}\\1&0&0&0&14\\0&1&0&0&0\\0&0&1&0&6\\0&0&0&1&16\\0&0&0&0&0\end{array}\right]$

So,

$A1=14\\A2=0\\B1=6\\B2=16$

From the above calculations, we see that

Factory A sent 14 loads of rice to Store 1

Factory A sent no loads of rice to Store 2

Factory B sent 6 loads of rice to Store 1

Factory B sent 16 loads of rice to Store 2

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