Question 1202250
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Part A


We focus on one elevator only.


x = number of adults
y = number of children
These variables are nonnegative integers 0,1,2,3,...


The elevator allows at most 12 people meaning that
{{{x+y <= 12}}}
Either x+y < 12, or x+y = 12


Now to the weight requirements.


Each adult weighs 175 pounds on average. 
Add on 10 pounds of their personal belongings to see that each adult contributes a weight of 175+10 = 185 pounds.
x number of adults contribute 185x pounds


Each child weighs 70 pounds on average. 
That bumps up to 80 pounds when considering their personal luggage. 
Therefore, y number of children adds on 80y extra pounds.


The total weight so far is 185x+80y pounds.
Then add on the 150 pounds of luggage mentioned in the instructions. 
This is the publicly shared luggage of the group (as opposed to the personal luggage).


Therefore, the total combined weight of each person and all luggage (personal or shared) is 185x+80y+150 pounds.


This total cannot exceed 1600 pounds. 
This is the ceiling value. 
Since we're dealing with an elevator, it is fitting to think of a hypothetical tower with 1600 floors as the highest we can go. 
Feel free to come up with other metaphors.


This leads to the inequality {{{185x+80y+150 <= 1600}}}
I'll let the student solve for y if the teacher requires it.


To recap everything we have these two inequalities
{{{system(x+y <= 12,185x+80y+150 <= 1600)}}}
that represent the system of inequalities


A more narrowly defined system would be
{{{system(x+y <= 12,185x+80y+150 <= 1600,x >= 0,y >= 0)}}}
which prevents x and y from being negative.
It makes no sense to have a negative number of people.


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Part B


Group 1 has 4 adults and 3 children.
x = 4
y = 3


Plug those into the 1st inequality
{{{x+y <= 12}}}
{{{4+3 <= 12}}}
{{{7 <= 12}}}
The 1st condition is satisfied.


Do the same for the other inequality
{{{185x+80y+150 <= 1600}}}
{{{185*4+80*3+150 <= 1600}}}
{{{1130 <= 1600}}}
The 2nd condition is satisfied as well.


Both conditions have been met. 
Group 1 can board the elevator.


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Group 2 has 1 adult and 11 children.
x = 1
y = 11


{{{x+y <= 12}}}
{{{1+11 <= 12}}}
{{{12 <= 12}}}
That works out


{{{185x+80y+150 <= 1600}}}
{{{185*1+80*12+150 <= 1600}}}
{{{1295 <= 1600}}}
That works as well.


Group 2 can board the elevator.

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Group 3 has 9 adults
x = 9
y = 0


{{{x+y <= 12}}}
{{{9+0 <= 12}}}
{{{9 <= 12}}}
That works out


{{{185x+80y+150 <= 1600}}}
{{{185*9+80*0+150 <= 1600}}}
{{{1815 <= 1600}}}
The last inequality is false, so condition #2 is NOT met


The extra weight is 1815-1600 = 215 pounds



Group 3 is NOT safely allowed to board the elevator.

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Summary:


Groups that can board a single elevator:
Group 1
Group 2


Group 3 is NOT safely allowed to board a single elevator.
Group 3 will have to split up in some way. At least two people must leave the elevator.
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