Question 321374
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.