Question 1172818
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This problem is a SPECIAL CASE: one equation is combined with one inequality.


The equation is

    x + y = 120.    (x for adults, y for children)


The inequality is

    500x + 400y <= 80000


Simplify this inequality by dividing both sides by 100

    5x + 4y <= 800.


So, you have, actually, this system (one equation and one inequality)

     x +  y  = 120

    5x + 4y <= 800


Also, the problem assumes that both quantities x and y are NON-NEGATIVE

    x >= 0,  y >= 0.


On the plot, it looks like this


    {{{graph(400,400,-20,200,-20,200,
             120-x, (800-5x)/4
)}}}


     Plot y = 120-x (red line) and y = {{{(800-5x)/4}}}  (green line)



The inequality represents all the points inside the triangle in QI under the green line.


The equation represents the red line.


So, the range for x is  0 <= x <= 120.

    The range for y is the same 0 <= y <= 120.


But x and y are not independent variables: they are interconnected by the equation x + y = 120.


This plot allows you to see the problem in whole and its solution, in particular.


The solution set is { the INTEGER points of the red line in QI }.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is the full analysis, &nbsp;presented in the form &nbsp;<U>AS &nbsp;IT &nbsp;SHOULD &nbsp;BE &nbsp;DONE</U>.