Question 30452
Big Video has a flat rate of $9.99/yr + $2.49/rentalo
Main Video has a flat rate of $20.49/yr + $1.79/rental
Let y = cost of renting videos
Let x = number of videos rented
the equations for the first year in each case are
{{{y = 9.99 + 2.49*x}}}
{{{y = 20.49 + 1.79*x}}}
substitute y in the first equation for y in the 2nd
{{{9.99 + 2.49*x = 20.49 + 1.79*x}}}
subtract 9.99 from both sides
{{{2.49*x = 10.50 + 1.79*x}}}
subtract 1.79*x from both sides
{{{.7*x = 10.50}}}
{{{x = 15}}}
What that says is, assuming that the rentals from both 
stores are all done in the first year, after 15 rentals 
from each, the cost of renting is the same at each store
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There are 4 possibilities, however
[1] rentals from both stores are in the same year
[2] rentals from both stores spill over into the next year
[3] rentals from Big Video only spill over into next year
[4] rental from Main St Video only spill over into next year
The statement of the problem sounds like the video rentals are
in the same time period, so I'll eliminate cases [3] & [4],
but to cover case [2], I write new equations
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{{{y = 19.98 + 2.49*x}}}
{{{y = 40.98 + 1.79*x}}}
I doubled the flat yearly cost for each store
{{{19.98 + 2.49*x = 40.98 + 1.79*x}}}
{{{2.49*x = 21.00 + 1.79*x}}}
{{{.7*x = 21.00}}}
{{{x = 30}}}
So it would take 30 rentals for the costs to be equal if all the 
rentals did not occur in the 1st year
I think it's kind of a trick question
Here are graphs that illustrate cases [1] & [2]
{{{graph(250,250,-5,20,-5,60, 9.99 + 2.49x,20.49 + 1.79x)}}} 
{{{graph(250,250,-5,40,-5,100, 19.98 + 2.49x,40.98 + 1.79x)}}}  
to check solutions, case [1] 
{{{y = 19.98 + 2.49*15}}}
{{{y = 9.99 + 37.35}}}
{{{y = 47.34}}}
{{{y = 20.49 + 1.79*15}}}
{{{y = 20.49 + 26.85}}}
{{{y = 47.34}}}
costs are equal as they should be
checking case [2]
{{{y = 19.98 + 2.49*30}}}
{{{y = 19.98 + 74.70}}}
{{{y = 94.68}}}
{{{y = 40.98 + 1.79*30}}}
{{{y = 40.98 + 53.70}}}
{{{y = 94.68}}}
costs are equal