Question 1000074
it took some doing, but after several false starts, i finally got it.

let x = how much his luggage weighed.
let y = how much her luggage weighed.
let a = how much weight is allowed free.
let c = the cost per pound for the excess weight.


presumably the cost per pound of the excess luggage is the same.


you get:


(x - a) * c = 1.5 (first equation)
(y - a) * c = 2.0 (second equation)
(x + y - a) * c = 13.5 (third equation)


add the first 2 equations together to get:


(x-a) * c + (y - a) * c = 1.5 + 2.0
simplify to get:
xc - ac + yc - ac = 3.5
combine like terms to get:
xc + yc - 2ac = 3.5


simplify the third equation to get:
xc + yc - ac = 13.5


you have 2 equations that can be solved simultaneously.


they are:


xc + yc - ac = 13.5
xc + yc - 2ac = 3.5


factor out the c in each equation to get:


(x + y - a) * c = 13.5
(x + y - 2a) * c = 3.5


since x + y = 94, you can replace x + y with 94 in these equations to get:


(94 - a) * c = 13.5
(94 - 2a) * c = 3.5


solve for c in both these equations to get:


c = 13.5 / (94 - a)
c = 3.5 / (94 - 2a)


subtract the second equation from the first to get:


0 = 13.5 / (94 - a) - 3.5 / (94 - 2a)
flip sides in the equation to get:
13.5 / (94 - a) - 3.5 / (94 - 2a) = 0


multiply both sides of this equation by (94 - a) * (94 - 2a) to get:


13.5 * (94 - 2a) - 3.5 * (94 - a) = 0


simplify to get:


13.5 * 94 - 13.5 * 2a - 3.5 * 94 + 3.5 * a = 0


simplify further to get:


1269 - 27a - 329 + 3.5a = 0


combine like terms to get:


940 - 23.5a = 0


add 23.5a to both sides of the equation to get:


940 = 23.5a


divide both sides of this equation by 23.5 and solve for a to get:


a = 40


that's your solution.


the allowed weight is 40 pounds.


now that you know that, you can solve for c.


your original 3 equations are, once again:


(x - a) * c = 1.5 (first equation)
(y - a) * c = 2.0 (second equation)
(x + y - a) * c = 13.5 (third equation)


when a = 40, these equations become:


(x - 40) * c = 1.5 (first equation)
(y - 40) * c = 2.0 (second equation)
(x + y - 40) * c = 13.5 (third equation)


add the first 2 equations together to get:


(x - 40) * c + (x - 40) * c = 3.5
simplify and combine like terms to get:
(x + y - 80) * c = 3.5


you now have two equations that can be solved simulteneously.


they are:


(x + y - 80) * c = 3.5
(x + y - 40) * c = 13.5


since x + y = 94, these equations become:


(94 - 80) * c = 3.5
(94 - 40) * c = 13.5


simplify to get:


14c = 3.5
54c = 13.5


solve for c in both equations to get:


c = 3.5/14 = .25
c = 13.5/54 = .25


c is equal to .25


you can now solve for the weight of his luggage and the weight of her luggage.


(x-a)*.25 = 1.5 becomes (x-40)*.25 = 1.5 which becomes .25x - .25*40 = 1.5 which becomes .25x - 10 = 1.5 which becomes .25x = 11.5 which gets you x = 11.5/.25 which gets you x = 46.


if the weight of his luggage is 46 pounds, then the weight of her luggage has to be 48 pounds because 94 = 46 + 48.


he's 6 pounds over.
she's 8 pounds over.


6*.25 = 1.5
8*.25 = 2.0


everything is confirmed as good.


your answer is that the allowed weight = 40 pounds.