Thread: Solving Systems by Matrix Reduction

1. Solving Systems by Matrix Reduction

Could someone kindly provide me with the process of how I can solve the following word problem using matrices?

Question: A firm produces three products, A, B and C that require processing by three machines, I, II, III. The time in hours required for processing one unit of each product by the three machines is given by the following table:

A B C
I 3 1 2

II 1 2 1

III 2 4 1

Machine I is available for 490 hours, machine II is available for 310 hours, and machine III for 560 hours. Find how many units of each product should be produced to make use of all the available time on the machines.

I am sorry, I don't know how to post a proper table for my question. But in the table above IA corresponds to entry 3, IB = 1, IC = 2, IIA=1, IIB=2, IIIC=1, IIIA=2, IIIB=4, IIIC=1

2. Hello, alisheraz19!

I'm not sure that matrices is the best way to go . . .

A firm produces three products: $A, B, C$
that require processing by three machines: $I, II, III$.
The time in hours required for processing one unit of each product
by the three machines is given by the following table:

. . $
\begin{array}{c||c|c|c||c|}
& A & B & C & \text{Total} \\ \hline\hline

I & 3 & 1 & 2 & 490 \\ \hline

II & 1 & 2 & 1 & 310 \\ \hline
III & 2 & 4 & 1 & 560 \\ \hline \end{array}$

Macine I is available for 490 hours,
machine II is available for 310 hours,
machine III for 560 hours.

Find how many units of each product should be produced
to make use of all the available time on the machines.
Let: . $\begin{Bmatrix} a &=& \text{number of A's} \\ b &=& \text{number of B's} \\ c &=& \text{number of C's} \end{Bmatrix}$

We have three inequalities: . $\begin{array}{ccc}a + b + 2c & \leq & 490 \\ a + 2b + c & \leq & 310 \\ 2a + 4b + c & \leq & 560 \end{array}$

Solve the system of equations: . $\begin{array}{ccc} 3a + b + 2c &{\color{red}=}& 490 \\ a + 2b + c &{\color{red}=}& 310 \\ 2a + 4b + c &{\color{red}=}& 560\end{array}$

. . and we get: . $\begin{Bmatrix}a\;=\;98 \\ b \;=\;76 \\ c \;=\;60\end{Bmatrix}$