document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1048965>Question 1048965</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi<BR>\n" );
document.write( "in a textbook the following problem was as follows.james has some matchsticks and sachets. If one matchstick is put into each sachet there are 2 matchsticks left.if he puts 2 matchsticks into each sachet he will have 24 sachets left. The solutions are m-s = 2 and s-m/2 = 24.\r<BR>\n" );
document.write( "\n" );
document.write( "Can u help me understand how these equations were derived.\r<BR>\n" );
document.write( "\n" );
document.write( "thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #664537 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=664537\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let m equal the number of matchsticks.<BR>\n" );
document.write( "let s equal the number of sachets.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "first verbal statement:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if one matchstick is put into each sachet, there are 2 matchsticks left.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in order for this to occur, you have to have 2 more matchsticks than sachets.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your equation is therefore m = s + 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "subtract s from both sides of this equation to get m - s = 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "second verbal statement:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if 2 matchsticks are put in each sachet, there are 24 sachets left.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in order for this to occur, you have to have 24 more sachets than half the number of marbles.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the equation for this is therefore s = m/2 + 24.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "subtract m/2 from both sides of this equation to get s - m/2 = 24.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "another way these can be derived which is a little more difficult to understand is as follows:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when you put one matchstick into each sachet, you are left with 2 matchsticks.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is like a division.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you are dividing the number of sachets by the number of matches and your result is 1 with a remainder of 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in a division, this is shown as s/m = 1 + 2/m\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the remainder is really the fraction of what's left over divided by the divisor.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so you start with s/m = 1 + 2/m<BR>\n" );
document.write( "then you multiply both sides of the equation by m to get s = m + 2<BR>\n" );
document.write( "then you subtract m from both sides of the equation to gets - m = 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the second one is a little trickier but should give you the correct result if you do it right.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the statement says that if you put 2 matchsticks in each sachet, then you have 24 sachets left.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this means that is you divide the number of sachets by the number of marbles, your result will be 1/2 + a remainder of 24 sachets.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the algebraic equation for this would be s/m = 1/2 + 24/m.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "multiply both sides of this equation by m and you get s = 1/2 * m + 24<BR>\n" );
document.write( "subtract 1/2 * m from both sides of this equation to get s - 1/2 * m = 24<BR>\n" );
document.write( "this can be shown as s - m/2 = 24.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "from these two equations, you can solve for the number of sachets and the number of marbles.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the two equations are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "m - s = 2<BR>\n" );
document.write( "s - m/2 = 24\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for m in the first equation to get m = s + 2<BR>\n" );
document.write( "replace m with s + 2 in the second equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "s - (s+2)/2 = 24\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "multiply both sides of this equation by 2 to get 2s - (s+2) = 48\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "simplify to get 2s - s - 2 = 48\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "combine like terms to get s - 2 = 48\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "add 2 to both sides of this equation to get s = 50\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since m = s + 2, then m = 52.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your solution is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "m = 52<BR>\n" );
document.write( "s = 50\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your first equation of m = s + 2 is satisfied because 52 = 50 + 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your second equation of s - m/2 = 24 is satisfied because 50 - 52/2 = 50 - 26 = 24.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "both equations are satisfied so the value of m must be 52 and the value of s must be 50.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "best i can do.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "hopefully it makes sense to you.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "first equations wasn't that hard to derive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "second equation took a little more thought.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );