document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1032212>Question 1032212</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Bethany’s credit card bill in December was 1.5 times her food bill. Her apartment rent was 5 times her utility bill. Her phone bill was $17 less than half of her food bill. Her phone bill was also $18 less than twice her utilities bill. The total of the five bills was $751. What was the amount of each bill? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #647030 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=647030\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your goal is to reduce everything down to one variable which can then be solved.<BR>\n" );
document.write( "once you solve for that one variable, you can then find the values of the other variables.\r<BR>\n" );
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document.write( "let:\r<BR>\n" );
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document.write( "a = apartment rent bill.<BR>\n" );
document.write( "c = credit card bill.<BR>\n" );
document.write( "p = phone bill.<BR>\n" );
document.write( "f = food bill.<BR>\n" );
document.write( "u = utility bill.\r<BR>\n" );
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document.write( "since the total bill is 751, you get:\r<BR>\n" );
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document.write( "a + c + p + f + u = 751.\r<BR>\n" );
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document.write( "now you start looking at relationships that will allow you to reduce the number of variables down to 1.\r<BR>\n" );
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document.write( "since c = 1.5 * f, and p = f/2 - 17, you can replace c and p with their equivalent values to get:\r<BR>\n" );
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document.write( "a + 1.5 * f + f/2 - 17 + f + u = 751.\r<BR>\n" );
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document.write( "since a = 5 * u, you can replace a to get:\r<BR>\n" );
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document.write( "5 * u + 1.5 * f + f/2 - 17 + f + u = 751.\r<BR>\n" );
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document.write( "you're down to 2 variables.\r<BR>\n" );
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document.write( "you are given that p = f/2 - 17 and also that p = 2u - 18.\r<BR>\n" );
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document.write( "in the second equation, you can solve for u to get u = (p + 18) / 2.\r<BR>\n" );
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document.write( "since you also know that p = f/2 - 17, you can replace p in the u = equation to get u = (f/2 - 17 + 18) / 2.\r<BR>\n" );
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document.write( "simplify this to get u = (f/2 + 1) / 2.\r<BR>\n" );
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document.write( "simplify this further to get u = f/4 + 1/2.\r<BR>\n" );
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document.write( "go back to the equation where you only had f and u.\r<BR>\n" );
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document.write( "that equation is 5 * u + 1.5 * f + f/2 - 17 + f + u = 751.\r<BR>\n" );
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document.write( "replace u with f/4 + 1/2 to get:\r<BR>\n" );
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document.write( "5 * (f/4 + 1/2) + 1.5 * f + f/2 - 17 + f + f/4 + 1/2 = 751.\r<BR>\n" );
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document.write( "simplify this equation to get:\r<BR>\n" );
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document.write( "5f/4 + 5/2 + 1.5f + f/2 - 17 + f + f/4 + 1/2 = 751.\r<BR>\n" );
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document.write( "multiply both sides of this equation by 4 to get:\r<BR>\n" );
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document.write( "5f + 10 + 6f + 2f - 68 + 4f + f + 2 = 3004.\r<BR>\n" );
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document.write( "that got rid of the fractions and makes combining like terms easier.\r<BR>\n" );
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document.write( "combine like terms to get 18f - 56 = 3004\r<BR>\n" );
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document.write( "add 56 to both sides of the equation to get 18f = 3060.\r<BR>\n" );
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document.write( "divide both sides of the equation by 18 to get f = 170.\r<BR>\n" );
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document.write( "now that you know f, you can solve for all the other variables.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "f = 170<BR>\n" );
document.write( "c = 255<BR>\n" );
document.write( "p = 68<BR>\n" );
document.write( "u = 43<BR>\n" );
document.write( "a = 215.\r<BR>\n" );
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document.write( "they add up to 751 so you're ok there.\r<BR>\n" );
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document.write( "the requirements of the problem are also satisfied.\r<BR>\n" );
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document.write( "255 / 170 = 1.5 so c = 1.5 * f is satisfied.<BR>\n" );
document.write( "5 * 43 = 215 so a = 5 * u is satisfied.<BR>\n" );
document.write( "68 = 1/2 * 170 - 17 so p = 1/2 * f - 17 is satisfied.<BR>\n" );
document.write( "68 = 2 * 43 - 18 so p = 2 * u - 18 is satisfied.\r<BR>\n" );
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document.write( "everything checks out so the answer is good.\r<BR>\n" );
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document.write( "the main objective was to reduce all the variables to one variable that could be solved.\r<BR>\n" );
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document.write( "once you did that you were able to solve for the rest of the variables.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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