document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=489628>Question 489628</A>: <I><SPAN STYLE=\"word-break: break-all;\"> given ac is = to df <BR>\n" );
document.write( "      bc is = to ef<BR>\n" );
document.write( "prove ab is = to de </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #333711 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=333711\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> presumably these are all part of the same line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you have the line of af equal to ab + bc + cd + de + ef\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the line would look like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         ---------------------------------------------------\r\n" );
document.write( "         a         b         c         d         e         f\r\n" );
document.write( "\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you are given that the line segment ac is equal to df.<BR>\n" );
document.write( "you are also given that the line segment bc is equal to ef\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that would look like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "\r\n" );
document.write( "         ---------------------------------------------------------\r\n" );
document.write( "         a         b         c               d         e         f\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "         ---------------------               ---------------------\r\n" );
document.write( "         a                   c     equals    d                   f\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "                   -----------                         -----------\r\n" );
document.write( "                   b         c         equals          e         f\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you are given that ac = df<BR>\n" );
document.write( "you know that ac = ab + bc<BR>\n" );
document.write( "you know that df = de + ef\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this means that ab + bc = de + ef by substitution of equals.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you are given that bc = ef\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this means that you can substitute bc for ef or you can substitute ef for bc in any equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the equation of interest is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ab + bc = de + ef\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "substitute bc for ef to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ab + bc = de + bc\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "subtract bc from both sides of this equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ab = de\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can do this because of the basic algebraic property that says:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if a + c = b + c then a = b\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this follows from the fact that if you subtract the same quantity from both sides of an equality, then the equality is preserved.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you are left with ab = de which is what you wanted to prove so you are done.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );