document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169567>Question 1169567</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are three options for fans purchasing a band's new release CD. They can purchase the CD, a premium CD bundle, or a deluxe CD bundle. A CD costs $7. A deluxe CD bundle costs the same as 4 CDs and one premium CD bundle. The band sells 317 CDs, 105 premium CD bundles, and 30 deluxe CD bundles for a total of $8,459. Find the price of each option.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #794619 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=794619\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "x = cost of a CD<BR>\n" );
document.write( "y = cost of a premium CD bundle<BR>\n" );
document.write( "4x+y = cost of a deluxe CD bundle<br><BR>\n" );
document.write( "The cost of 317 CD's, 105 premium bundles, and 30 deluxe bundles is $8,459:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=317%28x%29%2B105%28y%29%2B30%284x%2By%29+=+8459 ALIGN=MIDDLE  ALT=\"317%28x%29%2B105%28y%29%2B30%284x%2By%29+=+8459\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=317x%2B105y%2B120x%2B30y+=+8459 ALIGN=MIDDLE  ALT=\"317x%2B105y%2B120x%2B30y+=+8459\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=437x%2B135y+=+8459 ALIGN=MIDDLE  ALT=\"437x%2B135y+=+8459\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is a Diophantine equation: one equation with two unknowns, with the requirement that both variables be positive integers.<br><BR>\n" );
document.write( "Because of the size of the coefficients, it is likely that there is only one solution in positive integers.<br><BR>\n" );
document.write( "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><BR>\n" );
document.write( "Rewrite the equation as<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=135y+=+8459-437x ALIGN=MIDDLE  ALT=\"135y+=+8459-437x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%288459-437x%29%2F135 ALIGN=MIDDLE  ALT=\"y+=+%288459-437x%29%2F135\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "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><BR>\n" );
document.write( "That answer is x=7 and y=40.<br><BR>\n" );
document.write( "You can find that answer with pencil and paper, but the work is tedious.<br><BR>\n" );
document.write( "Starting with that last equation,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%288459-437x%29%2F135 ALIGN=MIDDLE  ALT=\"y+=+%288459-437x%29%2F135\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "simplify it by performing the division on the right to get a quotient and remainder:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%2862-3x%29%2B%2889-32x%29%2F135 ALIGN=MIDDLE  ALT=\"y+=+%2862-3x%29%2B%2889-32x%29%2F135\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then just try x=1, x=2, x=3, and so on until you find the x value that makes the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2889-32x%29%2F135 ALIGN=MIDDLE  ALT=\"%2889-32x%29%2F135\"   DISABLED_style= \"max-width:90%; height:auto;\" > an integer.<br><BR>\n" );
document.write( "ANSWERS:<BR>\n" );
document.write( "cost of a CD: x = $7<BR>\n" );
document.write( "cost of a premium CD bundle: y = $40<BR>\n" );
document.write( "cost of a deluxe CD bundle: 4x+y = $68<br><BR>\n" );
document.write( "CHECK: 317(7)+105(40)+30(68) = 2219+4200+2040 = 8459<br><BR>\n" );
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