document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104904>Question 1104904</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In one month a store sold pads of paper, some for $9, some for $2, and some for 50 cents. A total of 90 pads were sold for a total of $90. evaluate the total number of the 50 cent pads sold minus the number of $2 pads sold.<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 #719630 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</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=719630\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let x, y, and z be the numbers of $9, $2, and $0.50 pads sold.  Then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By%2Bz+=+90 ALIGN=MIDDLE  ALT=\"x%2By%2Bz+=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [the total number of pads sold was 90]<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%2B2y%2B.5z+=+90 ALIGN=MIDDLE  ALT=\"9x%2B2y%2B.5z+=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [the total cost of the pads sold was $90]<br><BR>\n" );
document.write( "This is a Diophantine equation -- three unknowns but only two equations; but we can find a solution using the fact that the unknowns have positive integer values.<br><BR>\n" );
document.write( "The general technique for solving a Diophantine system with three variables and two equations is to eliminate one of the variables to obtain a single equation in two variables; then analyze that equation to find integer solutions.<br><BR>\n" );
document.write( "You can choose any one of the variables to be the one to be eliminated.  The amount of work required to finish the problem can vary a great deal depending on which variable you eliminate first.  Unfortunately, it takes a huge amount of experience with solving this kind of problem to know ahead of time which variable is the best one to eliminate.<br><BR>\n" );
document.write( "So I will choose to eliminate variable z -- mostly because of the fractional coefficient on z in the second equation.  So double the second equation and then subtract one equation from the other:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By%2Bz+=+90 ALIGN=MIDDLE  ALT=\"x%2By%2Bz+=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x%2B4y%2Bz+=+180 ALIGN=MIDDLE  ALT=\"18x%2B4y%2Bz+=+180\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17x%2B3y+=+90 ALIGN=MIDDLE  ALT=\"17x%2B3y+=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Quick inspection of that equation shows that one solution is (x,y) = (0,30).  However, since the problem implies that some of each size pad were sold, we can reject that solution.<br><BR>\n" );
document.write( "To find other solutions in positive integers, since the coefficients 3 and 17 are relatively prime, we can take our rejected (0,30) solution and add 3 to x and subtract 17 from y, giving us (x,y) = (3,13).<br><BR>\n" );
document.write( "The next possible solution would be found by again adding 3 to x and subtractin 17 from y; but that would give us a negative value for y.<br><BR>\n" );
document.write( "So our only possible solution is (x,y) = (3,13).<br><BR>\n" );
document.write( "That gives us z = (90-(3+13)) = 74.<br><BR>\n" );
document.write( "So the store sold 3 $9 pads, 13 $2 pads, and 74 50-cent pads.<br><BR>\n" );
document.write( "CHECK: The total cost is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%289%29%2B13%282%29%2B74%28.5%29+=+27%2B26%2B37+=+90 ALIGN=MIDDLE  ALT=\"3%289%29%2B13%282%29%2B74%28.5%29+=+27%2B26%2B37+=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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