document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=177030>Question 177030</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello!<BR>\n" );
document.write( "How do I solve this determinant problem. <BR>\n" );
document.write( "I know the answer is 5/3 , -1 but am not sure how?<BR>\n" );
document.write( "Thanks for any help.\r<BR>\n" );
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document.write( "Solve<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C3%2Cabs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29%2C+%22=%22%2C+-3%29++%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "for x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #132082 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=132082\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Take note that the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is of the form <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=x ALIGN=MIDDLE  ALT=\"b=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-2 ALIGN=MIDDLE  ALT=\"c=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-x ALIGN=MIDDLE  ALT=\"d=-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=-3 ALIGN=MIDDLE  ALT=\"e=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=1 ALIGN=MIDDLE  ALT=\"f=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g=-6 ALIGN=MIDDLE  ALT=\"g=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=3 ALIGN=MIDDLE  ALT=\"i=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29=-3 ALIGN=MIDDLE  ALT=\"abs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-3 ALIGN=MIDDLE  ALT=\"d=-3\"   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=abs%28matrix%283%2C3%2Ca%2Cb%2Cc%2Cd%2Ce%2Cf%2Cg%2Ch%2Ci%29%29=a%28ei-fh%29-b%2A%28di-fg%29%2Bc%28dh-eg%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general 3x3 <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/Determinant#3-by-3_matrices\">determinant</a> formula.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%283%2C3%2C4%2Cx%2C-2%2C-x%2C-3%2C1%2C-6%2C2%2C3%29%29=%284%29%28%28-3%29%283%29-%281%29%282%29%29-%28x%29%28%28-x%29%283%29-%281%29%28-6%29%29%2B%28-2%29%28%28-x%29%282%29-%28-3%29%28-6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the given values (see above)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=%284%29%28%28-3%29%283%29-%281%29%282%29%29-%28x%29%28%28-x%29%283%29-%281%29%28-6%29%29%2B%28-2%29%28%28-x%29%282%29-%28-3%29%28-6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Replace the left side with \"d\" (see above)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=4%28-9-2%29-x%28-3x%2B6%29-2%28-2x-18%29 ALIGN=MIDDLE  ALT=\"d=4%28-9-2%29-x%28-3x%2B6%29-2%28-2x-18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=4%28-11%29-x%28-3x%2B6%29-2%28-2x-18%29 ALIGN=MIDDLE  ALT=\"d=4%28-11%29-x%28-3x%2B6%29-2%28-2x-18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-44-x%28-3x%2B6%29-2%28-2x-18%29 ALIGN=MIDDLE  ALT=\"d=-44-x%28-3x%2B6%29-2%28-2x-18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-44%2B3x%5E2-6x%2B4x%2B36 ALIGN=MIDDLE  ALT=\"d=-44%2B3x%5E2-6x%2B4x%2B36\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3=-44%2B3x%5E2-6x%2B4x%2B36 ALIGN=MIDDLE  ALT=\"-3=-44%2B3x%5E2-6x%2B4x%2B36\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-3 ALIGN=MIDDLE  ALT=\"d=-3\"   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=0=-44%2B3x%5E2-6x%2B4x%2B36%2B3 ALIGN=MIDDLE  ALT=\"0=-44%2B3x%5E2-6x%2B4x%2B36%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 3 to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=3x%5E2-2x-5 ALIGN=MIDDLE  ALT=\"0=3x%5E2-2x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "Notice we have a quadratic equation in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-5 ALIGN=MIDDLE  ALT=\"c=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%283%29%28-5%29+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%283%29%28-5%29+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-5 ALIGN=MIDDLE  ALT=\"c=-5\"   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=x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%283%29%28-5%29+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%283%29%28-5%29+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   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=x+=+%282+%2B-+sqrt%28+4-4%283%29%28-5%29+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4-4%283%29%28-5%29+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   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=x+=+%282+%2B-+sqrt%28+4--60+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4--60+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%283%29%28-5%29 ALIGN=MIDDLE  ALT=\"4%283%29%28-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-60 ALIGN=MIDDLE  ALT=\"-60\"   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=x+=+%282+%2B-+sqrt%28+4%2B60+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4%2B60+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284--60%29 ALIGN=MIDDLE  ALT=\"sqrt%284--60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%2B60%29 ALIGN=MIDDLE  ALT=\"sqrt%284%2B60%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=x+=+%282+%2B-+sqrt%28+64+%29%29%2F%282%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+64+%29%29%2F%282%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64 ALIGN=MIDDLE  ALT=\"64\"   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=x+=+%282+%2B-+sqrt%28+64+%29%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+64+%29%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   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=x+=+%282+%2B-+8%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+8%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64 ALIGN=MIDDLE  ALT=\"64\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   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=x+=+%282+%2B+8%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B+8%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+-+8%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%282+-+8%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2810%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%2810%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=++%28-6%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=++%28-6%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5%2F3 ALIGN=MIDDLE  ALT=\"x+=+5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1 ALIGN=MIDDLE  ALT=\"x+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. \r<BR>\n" );
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document.write( "So the answers are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5%2F3 ALIGN=MIDDLE  ALT=\"x+=+5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1 ALIGN=MIDDLE  ALT=\"x+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );