document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=282766>Question 282766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A teacher is buying supplies for the school. Using the tax-exempt status, she buys highlighters for 50 cents  each, three-ring binders for $3 each, and hole punchers for $10 each. She buys a total of 100 items and spent exactly $100. If she bought at least one of each item, the number of hole punchers she bought was:\r<BR>\n" );
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document.write( "(A)1 (B) 2 (C) 3 (D) 4 (E) 5\r<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 #205288 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=205288\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Your answer is that the number of hole punchers is 5.\r<BR>\n" );
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document.write( "You can't find it directly because you have 2 equations in 3 unknowns which means that one of the unknowns can only be found in relation to one of the other unknowns.\r<BR>\n" );
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document.write( "To find your solution, I did the following:\r<BR>\n" );
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document.write( "The two equations you have to solve simultaneously are:\r<BR>\n" );
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document.write( "x + y + z = 100<BR>\n" );
document.write( ".5x + 3y + 10z = 100\r<BR>\n" );
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document.write( "Multiply the second equation by 2 to get:\r<BR>\n" );
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document.write( "x + y + z = 100<BR>\n" );
document.write( "x + 6y + 20z = 200\r<BR>\n" );
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document.write( "Subtract first equation from second equation to get:\r<BR>\n" );
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document.write( "5y + 19z = 100\r<BR>\n" );
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document.write( "Solve y in terms of z to get:\r<BR>\n" );
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document.write( "y = (100 - 19z)/5\r<BR>\n" );
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document.write( "This is the same as:\r<BR>\n" );
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document.write( "y = 100/5 - 19z/5\r<BR>\n" );
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document.write( "y will not be an integer unless z = 5 or a multiple of 5 which means that the smallest z can be is 5.\r<BR>\n" );
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document.write( "Sounds like she could buy either 5 or 10 hole punchers and y would be an integer.\r<BR>\n" );
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document.write( "If z were 10, however, then she could not have bought any x or y because 10 * 10 = 100 which means zero money for x and y.\r<BR>\n" );
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document.write( "z would have to be equal to 5.\r<BR>\n" );
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document.write( "Here's how the equations work out.\r<BR>\n" );
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document.write( "x + y + 5 = 100<BR>\n" );
document.write( ".5x + 3y + 50 = 100\r<BR>\n" );
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document.write( "Subtract 5 from both sides of the first equation and subtract 50 from both sides of the second equation to get:\r<BR>\n" );
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document.write( "x + y = 95<BR>\n" );
document.write( ".5x + 3y = 50\r<BR>\n" );
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document.write( "Multiply the second equation by 2 to get\r<BR>\n" );
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document.write( "x + y = 95<BR>\n" );
document.write( "x + 6y = 100\r<BR>\n" );
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document.write( "Subtract the first equation from the second equation to get:\r<BR>\n" );
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document.write( "5y = 5\r<BR>\n" );
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document.write( "Divide both sides of the equation by 5 to get:\r<BR>\n" );
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document.write( "y = 1\r<BR>\n" );
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document.write( "Use the first equation to solve for x to get:\r<BR>\n" );
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document.write( "x = 94\r<BR>\n" );
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document.write( "Your values for x,y,z are 94,1,5.\r<BR>\n" );
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document.write( "Plug them into your original equations to get:\r<BR>\n" );
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document.write( "x + y + z = 100 becomes 94 + 1 + 5 = 100 which is true.<BR>\n" );
document.write( ".5x + 3y + 10z = 100 becomes .5*94 + 3*1 + 10*5 = 47 + 3 + 50 = 100 which is true.\r<BR>\n" );
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document.write( "Both original equations are true when you use the values for x,y,z of 94,1,5, so those values are good.\r<BR>\n" );
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document.write( "Your answer is:\r<BR>\n" );
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document.write( "She bought 5 hole punchers.\r<BR>\n" );
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document.write( "that would be selection E.\r<BR>\n" );
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