Question 1169567
<br>
x = cost of a CD
y = cost of a premium CD bundle
4x+y = cost of a deluxe CD bundle<br>
The cost of 317 CD's, 105 premium bundles, and 30 deluxe bundles is $8,459:<br>
{{{317(x)+105(y)+30(4x+y) = 8459}}}
{{{317x+105y+120x+30y = 8459}}}
{{{437x+135y = 8459}}}<br>
This is a Diophantine equation: one equation with two unknowns, with the requirement that both variables be positive integers.<br>
Because of the size of the coefficients, it is likely that there is only one solution in positive integers.<br>
The easiest way to find the solution(s) of an equation like this (with "large" coefficients) is with an excel spreadsheet or a graphing calculator.<br>
Rewrite the equation as<br>
{{{135y = 8459-437x}}}
{{{y = (8459-437x)/135}}}<br>
On a TI-83 calculator, enter that equation and use the table feature to find the only positive integer x value that yields a positive integer y value.<br>
That answer is x=7 and y=40.<br>
You can find that answer with pencil and paper, but the work is tedious.<br>
Starting with that last equation,<br>
{{{y = (8459-437x)/135}}}<br>
simplify it by performing the division on the right to get a quotient and remainder:<br>
{{{y = (62-3x)+(89-32x)/135}}}<br>
Then just try x=1, x=2, x=3, and so on until you find the x value that makes the remainder {{{(89-32x)/135}}} an integer.<br>
ANSWERS:
cost of a CD: x = $7
cost of a premium CD bundle: y = $40
cost of a deluxe CD bundle: 4x+y = $68<br>
CHECK: 317(7)+105(40)+30(68) = 2219+4200+2040 = 8459<br>