document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199298>Question 1199298</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A first number plus twice a second number is 13. Twice the first number plus the second totals 32. Find the numbers </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833117 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=833117\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "You have received two good responses showing typical algebraic solutions.  For this particular problem, because of the form of the given information, there is a different path to the solution that conceptually is much easier, although the formal algebra for this path is not more efficient.<br><BR>\n" );
document.write( "A first number plus twice a second number is 13. Twice the first number plus the second totals 32.<br><BR>\n" );
document.write( "Informally, using only mental arithmetic and logical reasoning, we can solve the problem like this:<br><BR>\n" );
document.write( "(1) Add the two pieces of given information to find that 3 times the first number plus 3 times the second number is 45.<BR>\n" );
document.write( "(2) From that, conclude that the sum of the two numbers is 15.<BR>\n" );
document.write( "(3) Compare (2) to the fact that the first number plus twice the second number is 13 to conclude that the second number is -2.<BR>\n" );
document.write( "(4) Use (2) and (3) to conclude that the first number is 17.<br><BR>\n" );
document.write( "ANSWERS: 17 and -2<br><BR>\n" );
document.write( "While that method of solving the problem informally is fast and easy, using formal algebra for that path to the solution is not very efficient....<br><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=13 ALIGN=MIDDLE  ALT=\"x%2B2y=13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By=32 ALIGN=MIDDLE  ALT=\"2x%2By=32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(3) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B3y=45 ALIGN=MIDDLE  ALT=\"3x%2B3y=45\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [add (1) and (2)]<BR>\n" );
document.write( "(4) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=15 ALIGN=MIDDLE  ALT=\"x%2By=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >    [divide by 3]<BR>\n" );
document.write( "(5) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2 ALIGN=MIDDLE  ALT=\"y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >      [subtract (4) from (1)]<BR>\n" );
document.write( "(6) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=17 ALIGN=MIDDLE  ALT=\"x=17\"   DISABLED_style= \"max-width:90%; height:auto;\" >      [from (4) and (5)]<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );