document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1105118>Question 1105118</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 $0.50. A total of 90 pads were sold for a total of $90. Evaluate the total number of the $0.50 pads sold minus the number of $2 pads sold. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #720061 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=720061\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x, y, and z be the numbers of $9, $2, and $0.50 pads sold, respectively.  Then<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;\" >  a total of 90 pads were sold<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 system of Diophantine equations: 3 variables but only 2 equations; but we can find the solution(s) using the fact that the variables have to have non-negative integer values.  And in this problem, since it says that all three types of pads were sold, the values in fact must be positive integers.<br><BR>\n" );
document.write( "(1) Eliminate one of the variables to get a system of one equation in two variables;<BR>\n" );
document.write( "(2) solve that equation for one variable in terms of the other; and<BR>\n" );
document.write( "(3) use the fact that the variable values must be positive integers to find the solution(s).<br><BR>\n" );
document.write( "We can double the second equation and subtract the two equations to eliminate z:<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( "We can see by inspection that (0,30) would be a solution if x could be 0; but we know x has to be a positive integer.<br><BR>\n" );
document.write( "With the coefficients 17 and 3 being relatively prime, all other potential solutions can be found starting with (0,30) and adding 3 to the value of x while subtracting 17 from the value of y.  We get<BR>\n" );
document.write( "(0,30), (3,13), (6,-4)...<BR>\n" );
document.write( "But the variable values have to be positive integers.<br><BR>\n" );
document.write( "So the only solution to this problem is x=3 and y=13, which makes z=74.<br><BR>\n" );
document.write( "CHECK: <BR>\n" );
document.write( "3+13+74 = 90  check<BR>\n" );
document.write( "3(9)+13(2)+74(.5) = 27+26+37 = 90  check </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );