A furniture factory has 2840 machine hours available each week in the cutting​ department, 3760 hours in the assembly​ department, and 2760 in the finishing department. Manufacturing a chair requires 0.9 hour of​ cutting, 0.6 hour of​ assembly, and 0.5 hour of finishing. A cabinet requires 0.2 hour of​ cutting, 0.8 hour of​ assembly, and 0.6 hour of finishing. A buffet requires 0.4 hour of​ cutting, 0.5 hour of​ assembly, and 0.3 hour of finishing. How many​ chairs, cabinets, and buffets should be produced in order to use all the available production​ capacity?

Answer: To use all available production capacity, 1,800 chairs, 2,100 cabinets and 2,000 buffets would be producedStep-by-step 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 be0.27x + 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 in0.45x + 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 get0.0224x = 40.32Therefore, x = 40.32 / 0.0224 = 1,800To 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 = 252Collect like terms0.18y = 252 + 1760.18y = 378Therefore, y = 378 / 0.18 = 2,100To 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 = 2,8402,040 + 0.4z = 2,840Collect like terms0.4z = 2,840 – 2,0400.4z = 800Therefore, z = 800 / 0.4 = 2,000From the above, it can be concluded that x (chairs) = 1,800 units, y (cabinets) = 2,100 units and z (buffets) = 2,000 units