SOLUTION: I cannot figure out how to do this!! Unit labor and material costs for manufacturing each of three types of products M, N, and P are given in the table below. Product

Question 321374: I cannot figure out how to do this!!
Unit labor and material costs for manufacturing each of three types of products M, N, and P are given in the table below.
Product
M N P
Labor 60 30 30
Materials 70 10 50
The weekly allocation for labor is $60,000 and for materials is $90,000. There are to be twice as many units of product M manufactured as units of product P. How many of each type of product should be manufactured each week to use exactly each of the weekly allocations? Hint: You will have three equations in three variables when you are done.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
As far as I can tell, there is no solution to this problem.

Here's why I think that way.

Your two simultaneous equations are:

60m + 30n + 30p = 60000 (labor equation)
70m + 10n + 50p = 90000 (material equation)

You are given that m = 2p

Based on that, you can substitute in both equations to eliminate the m variable.

Your equation become:

120p + 30n + 30p = 60000
140m + 10n + 50p = 90000

You combine like terms to ge t:

150p + 30n = 60000
190p + 10n = 90000

You multiply both sides of the second equation by 3 to get:

150p + 30n = 60000
570p + 30n = 270000

You subtract the first equation from the second equation to get:

420p = 210000

You divide both sides of the equation by 420 to get:

p = 500

You use the value of p to solve for n in both equations.

You get:

n = -500 from both equations.

In order for these 2 equations to be solved simultaneously, p has to be 500 and n has to be -500.

Since p = 500, m has to be equal to 1000 because m = 2p.

Your values are:

m = 1000
n = -500
p = 500

Your original equations are:

60m + 30n + 30p = 60000 (labor equation)
70m + 10n + 50p = 90000 (material equation)

substitute in these equations to get:

60*1000 + 30*(-500) + 30*500 = 60000 which is true.

and:

70*1000 + 10*(-500) + 50*100 = 90000 which is also true.

You have a solution but it's not a viable solution because n cannot be negative.

Your third equation was m = 2p which you used to substitute in the other 2 equations to eliminate m as a variable.

If you had put that in standard form, the equation would have been:

1m + 0n -2p = 0

Your 3 equations would then have been:

60m + 30n + 30p = 60000 (labor equation)
70m + 10n + 50p = 90000 (material equation)
1m + 0n -2p = 0 (m = 2p equation)

I double checked my work using mechanized linear equation solving tools and both tools that I used came up with the same answer which is the answer I provided you.

m = 1000
n = -500
p = 500

Again, this is not a valid solution because n cannot be negative.

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