document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206398>Question 1206398</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Bel is five times older than Tina.<BR>\n" );
document.write( "six years ago Bel was seven times older than Tina.<BR>\n" );
document.write( "what are their ages now? \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843842 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=843842\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the problem statement says:\r<BR>\n" );
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document.write( "Bel is 5 times older than Tina.<BR>\n" );
document.write( "6 years ago, <BR>\n" );
document.write( "Bel was 7 times older than Tina.<BR>\n" );
document.write( "what are their ages now.\r<BR>\n" );
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document.write( "a crucial phrase is \"5 times older\" and \"7 times older\".\r<BR>\n" );
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document.write( "this is not the same thing as \"5 times as old\" and \"7 times as old\".\r<BR>\n" );
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document.write( "if i am 2 times as old as you, then, if you are 10 years old, i am 20 years old.\r<BR>\n" );
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document.write( "if i am 2 times older than you, then, if you are 10 years old, i am 20 years older than you which makes my age 10 + 20 = 30.\r<BR>\n" );
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document.write( "i'll solve this problem with that interpretation to see where that leads us.\r<BR>\n" );
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document.write( "let y = bel's age and let x = tina's age.\r<BR>\n" );
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document.write( "the first equation becomes y = x + 5 * x which simplifies to y = 6 * x.\r<BR>\n" );
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document.write( "the second equation becomes y - 6 = (x - 6) + 7 * (x - 6) which becomes y - 6 = 8 * (x - 6).\r<BR>\n" );
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document.write( "solve for y in the second equation to get y = 8 * (x - 6) + 6\r<BR>\n" );
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document.write( "your 2 equations that need to be solved simultaneously are:\r<BR>\n" );
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document.write( "y = 6 * x<BR>\n" );
document.write( "y = 8 * (x - 6) + 6\r<BR>\n" );
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document.write( "simplify the second equation to get y = 8 * x - 48 + 6.<BR>\n" );
document.write( "this simplifies further to get y = 8 * x - 42\r<BR>\n" );
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document.write( "your 2 equations that need to be solved simultaneously become:\r<BR>\n" );
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document.write( "y = 6 * x<BR>\n" );
document.write( "y = 8 * x - 42\r<BR>\n" );
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document.write( "subtract the first equation from the second to get 0 = 2 * x - 42<BR>\n" );
document.write( "add 42 to both sides of that equati0on to get 42 = 2 * x<BR>\n" );
document.write( "solve for x to get x = 21\r<BR>\n" );
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document.write( "remember, y is bel's age and x is tina'a age, so this is saying that tina is 21 years old.\r<BR>\n" );
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document.write( "since y = 6 * x, then y = 126.<BR>\n" );
document.write( "since y is bel's age, then this is saying that bel is 126 years old.<BR>\n" );
document.write( "that's very old, but that's the way the math came out.\r<BR>\n" );
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document.write( "we now know that tina is 21 years old and bel is 26 years old.\r<BR>\n" );
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document.write( "the first problem statement says that bel is 5 times older than tina.<BR>\n" );
document.write( "tina is 21.<BR>\n" );
document.write( "5 * 21 = 105<BR>\n" );
document.write( "21 + 105 = 126.<BR>\n" );
document.write( "the first problem statement is confirmed as true.\r<BR>\n" );
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document.write( "the second problem statement says that, 6 years ago, bel was 7 times older than tina.<BR>\n" );
document.write( "6 years ago, bel was 126 - 6 = 120.<BR>\n" );
document.write( "6 years ago, tina was 21 - 6 = 15.<BR>\n" );
document.write( "this made bel 120 - 15 = 105 years older than tina.<BR>\n" );
document.write( "105 is equal to 7 * 15.<BR>\n" );
document.write( "the second problem statement is confirmed to be true.\r<BR>\n" );
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document.write( "you can solve these two equations graphically as shown below.<BR>\n" );
document.write( "the first equation is y = 6 * x<BR>\n" );
document.write( "the second equation is (y - 6) = 8 * (x - 6).<BR>\n" );
document.write( "the intersection of the 2 lines generate is at x = 21 and y = 126.\r<BR>\n" );
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document.write( "NOTE !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!\r<BR>\n" );
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document.write( "if the problem statement was:\r<BR>\n" );
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document.write( "Bel is five times as old as Tina.<BR>\n" );
document.write( "six years ago Bel was seven times as old as Tina.<BR>\n" );
document.write( "what are their ages now?\r<BR>\n" );
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document.write( "then you would get a different answer as shown below:\r<BR>\n" );
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document.write( "your 2 equations that need to be solved simultaneously become:\r<BR>\n" );
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document.write( "y = 5 * x<BR>\n" );
document.write( "y - 6 = 7 * (x - 6)\r<BR>\n" );
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document.write( "in the second equation, replace y with 5 * x to get:<BR>\n" );
document.write( "5 * x - 6 = 7 * (x - 6)<BR>\n" );
document.write( "simplify to get 5 * x - 6 = 7 * x - 42<BR>\n" );
document.write( "subtract 5 * x from both sides of this equation and add 42 to both sides of this equation to get 36 = 2 * x<BR>\n" );
document.write( "solve for x to get x = 18<BR>\n" );
document.write( "since y = 5 * x, then y = 90<BR>\n" );
document.write( "you have x = 18 and y = 90\r<BR>\n" );
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document.write( "tina is 18 and bel is 90.<BR>\n" );
document.write( "90 / 18 = 5.<BR>\n" );
document.write( "bel is 5 times as old as tina, confirming the first statement is correct.\r<BR>\n" );
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document.write( "6 years ago, bel is 84 and tina is 12.<BR>\n" );
document.write( "7 * 12 = 84, confirming the second statement is correct.\r<BR>\n" );
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document.write( "you have 2 answers.<BR>\n" );
document.write( "if the problem statement states \"older\", then tina is 21 and bel is 126.<BR>\n" );
document.write( "if the problem statement states \"as old as\", then tina is 18 and bel is 90.\r<BR>\n" );
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document.write( "bel being 90 is easier to accept than bel is 126.<BR>\n" );
document.write( "please check to make sure whether you meant \"older\" or \"as old as\".\r<BR>\n" );
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