document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1042192>Question 1042192</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Andrew has two children, David and Helen. The sum of their three ages is 49. David’s age is three times that of Helen. In 5 years time, Andrew’s age will be three times David’s age. What is the product of their ages now? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #657155 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=657155\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can solve this in several ways.<BR>\n" );
document.write( "i'll show you two.\r<BR>\n" );
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document.write( "the first way is by looking at this as three equations that need to be solved simultaneously through the elimination method so you can get a solution that is common to all three.\r<BR>\n" );
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document.write( "your variables will be:\r<BR>\n" );
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document.write( "a = andrew's age today.<BR>\n" );
document.write( "d = david's age today.<BR>\n" );
document.write( "h = helen's age today.\r<BR>\n" );
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document.write( "your first equation is a + d + h = 49\r<BR>\n" );
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document.write( "this equations says that the sum of their ages is 49.\r<BR>\n" );
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document.write( "your second equation is d = 3h\r<BR>\n" );
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document.write( "this equation says that david's age is 3 times that of helen's age.\r<BR>\n" );
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document.write( "your third equation is a + 5 = 3 * (d + 5)\r<BR>\n" );
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document.write( "this equation says that, in 5 years, andrew will be 3 times as old as david will be then.\r<BR>\n" );
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document.write( "andrew's age today is a.<BR>\n" );
document.write( "his age in 5 years is a + 5.<BR>\n" );
document.write( "david's age today is d.<BR>\n" );
document.write( "his age in 5 years is d + 5.\r<BR>\n" );
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document.write( "your three equations are:\r<BR>\n" );
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document.write( "a + d + h = 49<BR>\n" );
document.write( "d = 3h<BR>\n" );
document.write( "a + 5 = 3 * (d + 5)\r<BR>\n" );
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document.write( "rearrange the equations so that all the variable terms are on the left and all the constant terms are on the right.\r<BR>\n" );
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document.write( "your first equations stays the same.\r<BR>\n" );
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document.write( "your second equation starts as d = 3h.<BR>\n" );
document.write( "subtract 3h from both sides to get d - 3h = 0.\r<BR>\n" );
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document.write( "your third equation starts as a + 5 = 3 * (d + 5)<BR>\n" );
document.write( "simplify to get a + 5 = 3 * d + 3 * 5<BR>\n" );
document.write( "simplify further to get a + 5 = 3 * d + 15<BR>\n" );
document.write( "subtract 5 from both sides and subtract 3 * d from both sides to get a - 3 * d = 15 - 5<BR>\n" );
document.write( "simplify to get a - 3d = 10\r<BR>\n" );
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document.write( "your 3 equations are now:\r<BR>\n" );
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document.write( "a + d + h = 49 (equation 1)<BR>\n" );
document.write( "d - 3h = 0 (equation 2)<BR>\n" );
document.write( "a - 3d = 10 (equation 3)\r<BR>\n" );
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document.write( "subtract equation 2 from equation 1 to get:<BR>\n" );
document.write( "(a + d + h) - (d - 3h) = 49 - 0<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "a + d + h - d + 3h = 49<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "a + 4h = 49 (equation 4)\r<BR>\n" );
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document.write( "you eliminated d from equations 1 and 2 by subtracting equation 2 from equation 1.\r<BR>\n" );
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document.write( "now you want to eliminate d from equations 2 and 3.\r<BR>\n" );
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document.write( "equations 2 and 3 are:\r<BR>\n" );
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document.write( "d - 3h = 0 (equation 2)<BR>\n" );
document.write( "a - 3d = 10 (equation 3)\r<BR>\n" );
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document.write( "multiply both sides of equation 2 by 3 to get:\r<BR>\n" );
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document.write( "3d - 9h = 0 (equation 2a)<BR>\n" );
document.write( "a - 3d = 10 (equation 3)\r<BR>\n" );
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document.write( "add equations 2a and 3 together to get:\r<BR>\n" );
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document.write( "(3d - 9h) + (a - 3d) = 0 + 10<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "3d - 9h + a - 3d = 10<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "-9h + a = 10 (equation 5)\r<BR>\n" );
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document.write( "you have two equations that only have a and h in them.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "a + 4h = 49 (equation 4)<BR>\n" );
document.write( "-9h + a = 10 (equation 5)\r<BR>\n" );
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document.write( "you can eliminate the variable a by subtracting equation 5 from equation 4.<BR>\n" );
document.write( "subtract equation 5 from equation 4 to get:\r<BR>\n" );
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document.write( "(a + 4h) - (-9h + a) = 49 - 10<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "a + 4h + 9h - a = 39<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "13h = 39<BR>\n" );
document.write( "divide both sides of this equation by 13 to get:<BR>\n" );
document.write( "h = 3\r<BR>\n" );
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document.write( "you now know that the value of h is equal to 3.<BR>\n" );
document.write( "since d = 3h, then the value of d has to be 9.<BR>\n" );
document.write( "since a + d + h = 49, then the value of a has to be 37.\r<BR>\n" );
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document.write( "you have your solution.<BR>\n" );
document.write( "andrew is 37 years old.<BR>\n" );
document.write( "david is 9 years old.<BR>\n" );
document.write( "helen is 3 years old.\r<BR>\n" );
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document.write( "all the requirements of the problem are satisfied.<BR>\n" );
document.write( "the sum of their ages is 49.<BR>\n" );
document.write( "david is 3 times as old as helen.<BR>\n" );
document.write( "this is because 9 is equal to 3 * 3.<BR>\n" );
document.write( "in 5 years, andrew will be 3 times as old as david.<BR>\n" );
document.write( "this is because 37 + 5 = 42 and 9 + 4 = 14 and 42 = 3 * 14.\r<BR>\n" );
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document.write( "the above is the more formal way of solving this problem.<BR>\n" );
document.write( "the same method will be used for more complex problems as you get into solving simultaneous equations using matrices of coefficients and constants.\r<BR>\n" );
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document.write( "you can also solve it logically as follows:\r<BR>\n" );
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document.write( "start with your 3 original equations as before.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "a + d + h = 49 (first equation)<BR>\n" );
document.write( "d = 3h (second equation)<BR>\n" );
document.write( "a + 5 = 3 * (d + 5) (third equation)\r<BR>\n" );
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document.write( "since d = 3h, replace d in the first equation with 3h to get:\r<BR>\n" );
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document.write( "first equation becomes a + 3h + h = 49<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "first equation now equals a + 4h = 49\r<BR>\n" );
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document.write( "if you can get a in terms of h, then you can solve for h.\r<BR>\n" );
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document.write( "your third equation is a + 5 = 3 * (d + 5)<BR>\n" );
document.write( "simplify this to get a + 5 = 3 * d + 15<BR>\n" );
document.write( "subtract 5 from both sides of this equation to get a = 3 * d + 10<BR>\n" );
document.write( "since you already know that d = 3h, you can replace d with 3h to get:<BR>\n" );
document.write( "a = 3 * 3h + 10<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "a = 9h + 10\r<BR>\n" );
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document.write( "in your first equation of a + 4h = 49, replace a with (9h + 10) to get:<BR>\n" );
document.write( "(9h + 10) + 4h = 49<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "9h + 10 + 4h = 49<BR>\n" );
document.write( "combine like terms to get 13h + 10 = 49<BR>\n" );
document.write( "subtract 10 from both sides to get 13h = 39<BR>\n" );
document.write( "solve for h to get h = 3.\r<BR>\n" );
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document.write( "once you know h = 3, then d = 3h becomes 9 and a + d + h = 49 gets you a = 37.\r<BR>\n" );
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document.write( "same answer as before, only done through more informal type logic rather than a systematic analysis of solving three equations simultaneously using the elimination method.\r<BR>\n" );
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document.write( "this last method would fall more in line with solving three equations simultaneously through the substitution method.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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