document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1125614>Question 1125614</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Among all pairs of numbers (x,y) such that 4x+y=18, find the pair for which the sum of squares, x2+y2, is minimum.  Write your answers as fractions reduced to lowest terms. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #741910 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=741910\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "Solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4x%2By=18 ALIGN=MIDDLE  ALT=\"+4x%2By=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > for y\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4x%2By=18 ALIGN=MIDDLE  ALT=\"+4x%2By=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4x%2By-4x=18-4x ALIGN=MIDDLE  ALT=\"+4x%2By-4x=18-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 4x from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=18-4x ALIGN=MIDDLE  ALT=\"+y=18-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=-4x%2B18 ALIGN=MIDDLE  ALT=\"+y=-4x%2B18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now plug this into the function \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+x%5E2%2By%5E2 ALIGN=MIDDLE  ALT=\"z+=+x%5E2%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+x%5E2%2B%28y%29%5E2 ALIGN=MIDDLE  ALT=\"z+=+x%5E2%2B%28y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+x%5E2%2B%28-4x%2B18%29%5E2 ALIGN=MIDDLE  ALT=\"z+=+x%5E2%2B%28-4x%2B18%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice how y is replaced with -4x+18\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+x%5E2%2B%28-4x%2B18%29%28-4x%2B18%29 ALIGN=MIDDLE  ALT=\"z+=+x%5E2%2B%28-4x%2B18%29%28-4x%2B18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+x%5E2%2B16x%5E2-144x%2B324 ALIGN=MIDDLE  ALT=\"z+=+x%5E2%2B16x%5E2-144x%2B324\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+17x%5E2-144x%2B324 ALIGN=MIDDLE  ALT=\"z+=+17x%5E2-144x%2B324\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
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document.write( "The goal is to minimize z, so we want to find the smallest z value for some input x\r<BR>\n" );
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document.write( "The equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+17x%5E2-144x%2B324 ALIGN=MIDDLE  ALT=\"z+=+17x%5E2-144x%2B324\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"z+=+ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > with a = 17, b = -144, c = 324\r<BR>\n" );
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document.write( "The x coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"h+=+-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let's plug in the values for a and b.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+-%28-144%29%2F%282%2A17%29 ALIGN=MIDDLE  ALT=\"h+=+-%28-144%29%2F%282%2A17%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+144%2F34 ALIGN=MIDDLE  ALT=\"h+=+144%2F34\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+72%2F17 ALIGN=MIDDLE  ALT=\"h+=+72%2F17\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This is the x value that makes z the smallest. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+72%2F17 ALIGN=MIDDLE  ALT=\"x+=+72%2F17\"   DISABLED_style= \"max-width:90%; height:auto;\" > which leads to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-4x%2B18 ALIGN=MIDDLE  ALT=\"y+=+-4x%2B18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-4%2872%2F17%29%2B18 ALIGN=MIDDLE  ALT=\"y+=+-4%2872%2F17%29%2B18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-288%2F17%2B18 ALIGN=MIDDLE  ALT=\"y+=+-288%2F17%2B18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-288%2F17%2B306%2F17 ALIGN=MIDDLE  ALT=\"y+=+-288%2F17%2B306%2F17\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+18%2F17 ALIGN=MIDDLE  ALT=\"y+=+18%2F17\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the final answer is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+72%2F17 ALIGN=MIDDLE  ALT=\"x+=+72%2F17\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+18%2F17 ALIGN=MIDDLE  ALT=\"y+=+18%2F17\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This pair of values makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+x%5E2%2By%5E2 ALIGN=MIDDLE  ALT=\"z+=+x%5E2%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > the smallest possible, ie a minimum.<BR>\n" );
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