document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=113587>Question 113587</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help.\r<BR>\n" );
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document.write( "Find values for m and b in the following system so that the solution to the system is (-3, 4).\r<BR>\n" );
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document.write( "5x = 7y =b\r<BR>\n" );
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document.write( "mx + y = 22\r<BR>\n" );
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document.write( "Thanks<BR>\n" );
document.write( "JMS </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82654 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=82654\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "Find values for m and b in the following system so that the solution to the system is (-3, 4). \r<BR>\n" );
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document.write( "Given:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x+%2B+7y+=b ALIGN=MIDDLE  ALT=\"5x+%2B+7y+=b\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mx+%2B+y+=+22 ALIGN=MIDDLE  ALT=\"mx+%2B+y+=+22\"   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+=-3 ALIGN=MIDDLE  ALT=\"x+=-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=y+=+4 ALIGN=MIDDLE  ALT=\"y+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x+%2B+7y+=b ALIGN=MIDDLE  ALT=\"5x+%2B+7y+=b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%28-3%29+%2B+7%2A4+=b ALIGN=MIDDLE  ALT=\"5%28-3%29+%2B+7%2A4+=b\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15+%2B+28+=b ALIGN=MIDDLE  ALT=\"-15+%2B+28+=b\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13+=b ALIGN=MIDDLE  ALT=\"13+=b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+13 ALIGN=MIDDLE  ALT=\"b+=+13\"   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=mx+%2B+y+=+22 ALIGN=MIDDLE  ALT=\"mx+%2B+y+=+22\"   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=-3m+%2B+4+=+22 ALIGN=MIDDLE  ALT=\"-3m+%2B+4+=+22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "	<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3m+=+22+-+4 ALIGN=MIDDLE  ALT=\"-3m+=+22+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "	<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3m+=++18 ALIGN=MIDDLE  ALT=\"-3m+=++18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "	<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=++18%2F-3 ALIGN=MIDDLE  ALT=\"m+=++18%2F-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "	<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=++-+6 ALIGN=MIDDLE  ALT=\"m+=++-+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then we have system:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x+%2B+7y+=+13 ALIGN=MIDDLE  ALT=\"5x+%2B+7y+=+13\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6x+%2B+y+=+22 ALIGN=MIDDLE  ALT=\"-6x+%2B+y+=+22\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and its solution is:\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solving_linear_system_by_elimination.solver>Solving a System of Linear Equations by Elimination/Addition</A></TD></TR>\n" );
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document.write( "  <BR>\n" );
document.write( "  Lets start with the given system of linear equations<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B7%2Ay=13 ALIGN=MIDDLE  ALT=\"5%2Ax%2B7%2Ay=13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%2Ax%2B1%2Ay=22 ALIGN=MIDDLE  ALT=\"-6%2Ax%2B1%2Ay=22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 5 and -6 to some equal number, we could try to get them to the LCM.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Since the LCM of 5 and -6 is -30, we need to multiply both sides of the top equation by -6 and multiply both sides of the bottom equation by -5 like this:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%2A%285%2Ax%2B7%2Ay%29=%2813%29%2A-6 ALIGN=MIDDLE  ALT=\"-6%2A%285%2Ax%2B7%2Ay%29=%2813%29%2A-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the top equation (both sides) by -6<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2A%28-6%2Ax%2B1%2Ay%29=%2822%29%2A-5 ALIGN=MIDDLE  ALT=\"-5%2A%28-6%2Ax%2B1%2Ay%29=%2822%29%2A-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by -5<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So after multiplying we get this:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-30%2Ax-42%2Ay=-78 ALIGN=MIDDLE  ALT=\"-30%2Ax-42%2Ay=-78\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2Ax-5%2Ay=-110 ALIGN=MIDDLE  ALT=\"30%2Ax-5%2Ay=-110\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Notice how -30 and 30 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-30%2B30=0 ALIGN=MIDDLE  ALT=\"-30%2B30=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now add the equations together. In order to add 2 equations, group like terms and combine them<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-30%2Ax%2B30%2Ax%29-42%2Ay-5%2Ay%29=-78-110 ALIGN=MIDDLE  ALT=\"%28-30%2Ax%2B30%2Ax%29-42%2Ay-5%2Ay%29=-78-110\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-30%2B30%29%2Ax-42-5%29y=-78-110 ALIGN=MIDDLE  ALT=\"%28-30%2B30%29%2Ax-42-5%29y=-78-110\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28-30%2B30%29%2Ax%2B%28-42-5%29%2Ay=-78-110 ALIGN=MIDDLE  ALT=\"cross%28-30%2B30%29%2Ax%2B%28-42-5%29%2Ay=-78-110\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So after adding and canceling out the x terms we're left with:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-47%2Ay=-188 ALIGN=MIDDLE  ALT=\"-47%2Ay=-188\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-188%2F-47 ALIGN=MIDDLE  ALT=\"y=-188%2F-47\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-47 ALIGN=MIDDLE  ALT=\"-47\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for y<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4 ALIGN=MIDDLE  ALT=\"y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now plug this answer into the top equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B7%2Ay=13 ALIGN=MIDDLE  ALT=\"5%2Ax%2B7%2Ay=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for x<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B7%284%29=13 ALIGN=MIDDLE  ALT=\"5%2Ax%2B7%284%29=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4 ALIGN=MIDDLE  ALT=\"y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B28=13 ALIGN=MIDDLE  ALT=\"5%2Ax%2B28=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax=13-28 ALIGN=MIDDLE  ALT=\"5%2Ax=13-28\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28 ALIGN=MIDDLE  ALT=\"28\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax=-15 ALIGN=MIDDLE  ALT=\"5%2Ax=-15\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28%281%2F5%29%285%29%29%2Ax=%28-15%29%281%2F5%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F5%29%285%29%29%2Ax=%28-15%29%281%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F5 ALIGN=MIDDLE  ALT=\"1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cancel out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the terms on the right side<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So our answer is <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4 ALIGN=MIDDLE  ALT=\"y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  which also looks like<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Notice if we graph the equations (if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver</a>)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B7%2Ay=13 ALIGN=MIDDLE  ALT=\"5%2Ax%2B7%2Ay=13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%2Ax%2B1%2Ay=22 ALIGN=MIDDLE  ALT=\"-6%2Ax%2B1%2Ay=22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  we get <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++drawing%28+500%2C+600%2C+-13%2C+13%2C+-14%2C+14%2C%0D%0A++++graph%28+500%2C+600%2C+-13%2C+13%2C+-14%2C+14%2C+%2813-5%2Ax%29%2F7%2C+%2822--6%2Ax%29%2F1+%29%2C%0D%0A++++blue%28++circle%28+-3%2C+4%2C+0.173333333333333+%29+%29+%0D%0A++%29%0D%0A++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B7%2Ay=13 ALIGN=MIDDLE  ALT=\"5%2Ax%2B7%2Ay=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%2Ax%2B1%2Ay=22 ALIGN=MIDDLE  ALT=\"-6%2Ax%2B1%2Ay=22\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  and we can see that the two equations intersect at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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