Let the firm has
330 units of R1, 455 units of R2 and 140 units of R3 available with it and let the amount
of raw materials R1, R2 and R3 required to manufacture each unit of the three products
is given by
1
2
3
1
2
3
R
R
R
3
4
0
P
B
P
7
9
3
P
5
12
7




=






How many units of each product is to be made so as to utilise the full available raw
material Solution Step 1 The situation is easily identifiable Step 2 Suppose the firm produces x units of P1, y units of P2 and z units of P3 Since
product P1 requires 3 units of R1, P2 requires 7 units of R1 and P3 requires 5 units of R1
(observe matrix B) and the total number of units, of R1, available is 330, we have
3x + 7y + 5z = 330 (for raw material R1)
Similarly, we have
4x + 9y + 12z = 455 (for raw material R2)
and
3y + 7z = 140 (for raw material R3)
This system of equations can be expressed in matrix form as
3
7
5
4
9
12
0
3
7
330
455
140















=








x
y
z
Rationalised 2023-24
202
MATHEMATICS
Step 3 Using elementary row operations, we obtain
1
0
0
0
1
0
0
0
1
20
35
5















=








x
y
z
This gives x = 20, y = 35 and z = 5