Answer: To use all available production capacity, 1,800 chairs, 2,100 cabinets and 2,000 buffets would be produced
Explanation:
The best way to solve this is by using simultaneous equations. To do this, represent the products with variables. Say, X to represent chairs, y to represent cabinets and z to represent buffets.
From the information above, it can be solved thus:
0.9x + 0.2y + 0.4z = 2,840 _ eq (1)
0.6x + 0.8y + 0.5z = 3,760 _ eq (2)
0.5x + 0.6y + 0.3z = 2,760 _ eq (3)
To eliminate one of the variables, take two equations and multiply one by the coefficient of z in the other as follows:
Taking eq (1) and eq (3), multiply eq (1) by 0.3 to get eq (4) and eq (3) by 0.4 to get eq (5). The result should be
0.27x + 0.06y + 0.12z = 852 _ eq (4)
0.2x + 0.24y + 0.12z = 1,104 _ eq (5)
Then subtract eq (4) from eq (5) to get eq (6) as follows:
-0.07x + 0.18y = 252 _ eq (6)
Repeat the whole process again but this time using different combination of equations.
Taking eq (1) and eq (2), multiply eq (1) by 0.5 to get eq (7) and eq (2) by 0.4 to get eq (8). This should result in
0.45x + 0.1y + 0.2z = 1,420 _ eq (7)
0.24x + 0.32y + 0.2z = 1,594 _ eq (8)
Again, subtract eq (7) from eq (8) to get eq (9) as follows:
-0.21x + 0.22y = 84 _ eq (9)
To get the value of x, eliminate y by multiplying eq (6) by 0.22 and also eq (9) by 0.18. The result should be thus:
-0.0154x + 0.0396y = 55.44 _ eq (10)
-0.0378x + 0.0396y = 15.12 _ eq (11)
Subtract eq (11) from eq (10) to get
0.0224x = 40.32
Therefore, x = 40.32 / 0.0224 = 1,800
To get the value of y, substitute x = 1,800 into eq (6) or eq (9)
-0.07 (1,800) + 0.18y = 252 _ eq (6)
-176 + 0.18y = 252
Collect like terms
0.18y = 252 + 176
0.18y = 378
Therefore, y = 378 / 0.18 = 2,100
To get the value of z, substitute the values of x and y into any of eq (1), (2) or (3)
0.9(1,800) + 0.2(2,100) + 0.4z = 2,840 _ eq(1)
1,620 + 420 + 0.4z = 2,840
2,040 + 0.4z = 2,840
Collect like terms
0.4z = 2,840 – 2,040
0.4z = 800
Therefore, z = 800 / 0.4 = 2,000
From the above, it can be concluded that x (chairs) = 1,800 units, y (cabinets) = 2,100 units and z (buffets) = 2,000 units