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Hey, I have a question about the formulas. What I don’t get is why you change the numbers of the formulas and what the logic is, an example I have come across: (an instance of working out shadow prices)
Labour: 2x + 4Y = xxxxx
Materials: 4X + 2Y = xxxxxxAnswer: 4X + 8Y – how do you get to this, all the examples I have done are different so I am so confused as to how you change this? If this is incorrect, the whole formula won’t work so although I can complete the rest of the question fine, I can’t work out the logic to this part, can someone help? Much Appreciated.
I’ll explain this by putting some numbers in the equations so it’s hopefully clearer.
Labour: 2X + 4Y = 192
Materials: 4X + 2Y = 300What you need to do at the end is take away one equation from the other and end up with only one unknown, either X or Y. In this example the labour equation was multiplied by 2 so the labour equation became (2 x 2)X + (4 x 2)Y = 192 x 2 or 4X + 8Y = 384 so they’ve got the multiple of X in both equations to be the same (4X). . The two equations are now as below and you can see if you take away one from the other the X variable disappears.
Labour: 4X + 8Y = 384
Materials: 4X + 2Y = 300Labour – Materials
(4X + 8Y) – (4X + 2Y) = 384 – 300
4X – 4X + 8Y – 2Y = 84
0X + 6Y = 84
Y = 84 / 6 = 14″In this case it’s easy to spot that you can multiply or divide either of the equations by 2 and you’ll then be able to take one equation from the other to get an equation with just one unknown.
If you have more complicated figures it might not be as easy to see what to multiply or divide the equations by so you could use a slightly longer method that will always work. The labour equation has 2X so divide this equation by 2 so that it become X. The materials equation has 4X so divide this equation by 4 so that it also becomes X. Basically, whatever the multiple of X is, you should divide the whole equation by this amount.
Labour: 2X + 4Y = 192
Labour: X + ( 4 / 2 ) x Y = 192 / 2
Labour: X + 2Y = 96Materials: 4X + 2Y = 300
Materials: X + ( 2 / 4 ) x Y = 300 / 4
Materials: X + 0.5Y = 75Now, as above you can take one equation away from the other and end up with just one unknown
Labour – Materials
(X + 2Y) – (X + 0.5Y) = 96 – 75
X – X + 2Y – 0.5Y = 21
0X + 1.5Y = 21
Y = 21 / 1.5 = 14As you see the answer is the same, however you choose to multiply the equations you’ll end up with the same solution. In this case Y = 14 which you can then put back into one of the equations to calculate X.
Labour: 2X + 4Y = 192
Labour: 2X + 4 x 14 = 192
Labour: 2X + 56 = 192
Labour: X = ( 192 – 56 ) / 2 = 68Furgus: Please do not answer questions in this forum – it is the Ask the Tutor Forum, and you are not the tutor. (But please do help people in the other forum 🙂 )
Cassiemac89: What Furgus has written is correct. There are many ways that you can solve two equations together (obviously all giving the same final answer). You should have done this at school and so if you were taught a different way (and remember it!) then by all means do it that way.
Otherwise, I do show in detail what I think it is the easiest way in my free lectures on linear programming.
The lectures are a complete free course for Paper PM and cover everything needed to be able to pass the exam well.
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