document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210612>Question 1210612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Fill in the blanks to make an equation that is linear in disguise, and has a solution.<BR>\n" );
document.write( "sqrt(2x - 3) + __x + ___ = ___x - ___ + ____<BR>\n" );
document.write( "8 9 10 11 12 13 14 15 16 17 18 19 20  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854412 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5362)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854412\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The way I interpret the question, only the numbers given below the equation can be used to fill the blanks.<BR>\n" );
document.write( "I am going to refer to the values to be filled as the constants A, B, C, D and E.<BR>\n" );
document.write( "I will write the equation as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x+-+3%29+%2B+Ax+%2B+B+=+Cx+-+D+%2B+E ALIGN=MIDDLE  ALT=\"sqrt%282x+-+3%29+%2B+Ax+%2B+B+=+Cx+-+D+%2B+E\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It is not specified, but I am going to assume that each number can only be used once.<BR>\n" );
document.write( "A linear equation is one where the variable appears multiplied by some coefficient(s), but never with an exponent, or in a square root.<BR>\n" );
document.write( "The square root shown appears to make that equation not linear, so<BR>\n" );
document.write( "I am also going to assume, that the equation to be made is to be \"linear in disguise\" because<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x-3%29=0 ALIGN=MIDDLE  ALT=\"sqrt%282x-3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > that satisfies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax+%2B+B+=+Cx+-+D+%2B+E ALIGN=MIDDLE  ALT=\"Ax+%2B+B+=+Cx+-+D+%2B+E\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x-3%29=0 ALIGN=MIDDLE  ALT=\"sqrt%282x-3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3=0 ALIGN=MIDDLE  ALT=\"2x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2F2 ALIGN=MIDDLE  ALT=\"x=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2F2 ALIGN=MIDDLE  ALT=\"x=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a solution of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax+%2B+B+=+Cx+-+D+%2B+E ALIGN=MIDDLE  ALT=\"Ax+%2B+B+=+Cx+-+D+%2B+E\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the values of the constants must be such that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F2%29A%2BB+=%283%2F2%29C-D%2BE ALIGN=MIDDLE  ALT=\"%283%2F2%29A%2BB+=%283%2F2%29C-D%2BE\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3A%2B2B=3C-2D%2B2E ALIGN=MIDDLE  ALT=\"3A%2B2B=3C-2D%2B2E\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3A-3C=2E-2D-2B ALIGN=MIDDLE  ALT=\"3A-3C=2E-2D-2B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28A-C%29=2%28E-D-B%29 ALIGN=MIDDLE  ALT=\"3%28A-C%29=2%28E-D-B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The values for A, B, C, D, and E must be all different, be in the list of numbers given, and the numbers given are such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cabs%28A-C%29%3C=20-8=12 ALIGN=MIDDLE  ALT=\"0%3Cabs%28A-C%29%3C=20-8=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28A-C%29=2%28E-D-B%29 ALIGN=MIDDLE  ALT=\"3%28A-C%29=2%28E-D-B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A-C=2%28E-D-B%29%2F3 ALIGN=MIDDLE  ALT=\"A-C=2%28E-D-B%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  tells us that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A-C ALIGN=MIDDLE  ALT=\"A-C\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be a multiple of 2, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E-D-B ALIGN=MIDDLE  ALT=\"E-D-B\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be a multiple of 3.<BR>\n" );
document.write( "From there on, we can guess and check to find suitable answers. There is a very large number of them.<BR>\n" );
document.write( "For example, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28E=20%2CD=8%2CB=9%29 ALIGN=MIDDLE  ALT=\"system%28E=20%2CD=8%2CB=9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28E=20%2CD=9%2CB=8%29 ALIGN=MIDDLE  ALT=\"system%28E=20%2CD=9%2CB=8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > give us<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E-D-B=20-8-9=3 ALIGN=MIDDLE  ALT=\"E-D-B=20-8-9=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and are part of some of the many possible answers.<BR>\n" );
document.write( "Those choices make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A-C=2%2A3%2F3=2 ALIGN=MIDDLE  ALT=\"A-C=2%2A3%2F3=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and leaves us the numbers 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19 available to use as A and C.<BR>\n" );
document.write( "Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=12%2CC=10%29 ALIGN=MIDDLE  ALT=\"system%28A=12%2CC=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=13%2CC=11%29 ALIGN=MIDDLE  ALT=\"system%28A=13%2CC=11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=14%2CC=12%29 ALIGN=MIDDLE  ALT=\"system%28A=14%2CC=12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=15%2CC=13%29 ALIGN=MIDDLE  ALT=\"system%28A=15%2CC=13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=16%2CC=14%29 ALIGN=MIDDLE  ALT=\"system%28A=16%2CC=14%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=17%2CC=15%29 ALIGN=MIDDLE  ALT=\"system%28A=17%2CC=15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=18%2CC=16%29 ALIGN=MIDDLE  ALT=\"system%28A=18%2CC=16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=19%2CC=17%29 ALIGN=MIDDLE  ALT=\"system%28A=19%2CC=17%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > could be used to complete an answer.<BR>\n" );
document.write( "There are many other sets of values that are valid answers.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Let's verify if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=12%2CB=8%2CC=10%2CD=9%2CE=20%29 ALIGN=MIDDLE  ALT=\"system%28A=12%2CB=8%2CC=10%2CD=9%2CE=20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x-3%29%2B12x%2B8=+10x-9%2B20 ALIGN=MIDDLE  ALT=\"sqrt%282x-3%29%2B12x%2B8=+10x-9%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > an equation \"linear in disguise\" that  has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2F2 ALIGN=MIDDLE  ALT=\"x=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a solution.<BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2F2 ALIGN=MIDDLE  ALT=\"x=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x-3%29%2B12x%2B8=+10x-9%2B20 ALIGN=MIDDLE  ALT=\"sqrt%282x-3%29%2B12x%2B8=+10x-9%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > , I get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%283%2F2%29-3%29%2B12%283%2F2%29%2B8=10%283%2F2%29-9%2B20 ALIGN=MIDDLE  ALT=\"sqrt%282%283%2F2%29-3%29%2B12%283%2F2%29%2B8=10%283%2F2%29-9%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283-3%29%2B18%2B8=15-9%2B20 ALIGN=MIDDLE  ALT=\"sqrt%283-3%29%2B18%2B8=15-9%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%280%29%2B26=6%2B20 ALIGN=MIDDLE  ALT=\"sqrt%280%29%2B26=6%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2B26=26 ALIGN=MIDDLE  ALT=\"0%2B26=26\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=26=26 ALIGN=MIDDLE  ALT=\"26=26\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "For a more systematic approach, we could start with the possible values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A-C ALIGN=MIDDLE  ALT=\"A-C\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Knowing that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cabs%28A-C%29%3C=20-8=12 ALIGN=MIDDLE  ALT=\"0%3Cabs%28A-C%29%3C=20-8=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A-C ALIGN=MIDDLE  ALT=\"A-C\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not zero,<BR>\n" );
document.write( "the possible vales for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A-C ALIGN=MIDDLE  ALT=\"A-C\"   DISABLED_style= \"max-width:90%; height:auto;\" > are -12, -10, -8, -6, -4, -2, 2, 4, 6, 8, 10, and 12.<BR>\n" );
document.write( "That makes the possible values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E-B-D=3%28A-C%29%2F2 ALIGN=MIDDLE  ALT=\"E-B-D=3%28A-C%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > -18,-15,-12, -9, -6 -3, 3, 6, 9, 12, 15, and 18. <BR>\n" );
document.write( "To get a longer list of possible solutions, we could start by using multiples of 3 for E, B, and D to make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E-B-D ALIGN=MIDDLE  ALT=\"E-B-D\"   DISABLED_style= \"max-width:90%; height:auto;\" > a multiple of 3.<BR>\n" );
document.write( "Thinking about modular arithmetic, for those familiar with it, we could make a very long list of options for E, B, and D that make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E-B-D ALIGN=MIDDLE  ALT=\"E-B-D\"   DISABLED_style= \"max-width:90%; height:auto;\" > a multiple of 3<BR>\n" );
document.write( "Then, we could weed out from those lists those with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E-B-D=0 ALIGN=MIDDLE  ALT=\"E-B-D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and those with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28E-B-D%29%3E18 ALIGN=MIDDLE  ALT=\"abs%28E-B-D%29%3E18\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then we could find possible values of A and C, eliminating the ones where A or C was already used as E, B, or D.<BR>\n" );
document.write( "We are still left with hundreds of different valid answers. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );