document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=955337>Question 955337</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please Help me work this out step by step... I Haven't done a mixture problem with so many variables before. I appreciate all the help I can get<BR>\n" );
document.write( "Ethonal and Toluene are two examples of additives within your gasoline.<BR>\n" );
document.write( "Type A gasoline is 5% Ethanol and 10% Toluene<BR>\n" );
document.write( "Type B gasoline is 15% Ethanol and 8% Toluene <BR>\n" );
document.write( "Type C gasoline is 10% Ethanol and 4% Toluene <BR>\n" );
document.write( "A special Type D gasoline is 8.05% Ethanol and 6.84% Toluene. Precisely blending the correct quantities of Type A, B, and C together can create type D.<BR>\n" );
document.write( "How much of each type should be blended together to create 100 gallons of Type D gasoline.<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 #583589 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=583589\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let a = number of gallons of type A gasoline.<BR>\n" );
document.write( "let b = number of gallons of type B gasoline.<BR>\n" );
document.write( "let c = number of gallons of type C gasoline.<BR>\n" );
document.write( "let d = number of gallons of type D gasoline.\r<BR>\n" );
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document.write( "D = 100 gallons.\r<BR>\n" );
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document.write( "you have 3 equations that need to be solved simultaneously.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "a + b + c = 100\r<BR>\n" );
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document.write( "this equation tells you that the number of gallons of type A and type B and type C must be equal to the number of gallons of type D which is equal to 100 gallons.\r<BR>\n" );
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document.write( ".05a + .15b + .10c = .0805d = .0805 * 100 = 8.05\r<BR>\n" );
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document.write( "this equation tells you that the number of gallons of ethanol in type A and type B and type C gasoline must be equal to the number of gallons of ethanol in type D gasoline.\r<BR>\n" );
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document.write( ".10a + .08b + .04c = .0684d = .0684 * 100 = 6.84\r<BR>\n" );
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document.write( "this equation tells you that the number of gallons of toluene in type A and type B and type C gasoline must be equal to the number of gallons of toluene in type D gasoline.\r<BR>\n" );
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document.write( "the 3 equations are:\r<BR>\n" );
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document.write( "a + b + c = 100<BR>\n" );
document.write( ".05a + .15b + .10c = 8.05<BR>\n" );
document.write( ".10a + .08b + .04c = 6.84\r<BR>\n" );
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document.write( "when you solve these equations simultaneously, you will get:\r<BR>\n" );
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document.write( "a = 44<BR>\n" );
document.write( "b = 5<BR>\n" );
document.write( "c = 51\r<BR>\n" );
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document.write( "you will need 44 gallons of type A gasoline and 5 gallons of type B gasoline and 51 gallons of type C gasoline to make 100 gallons of type D gasoline.\r<BR>\n" );
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document.write( "solving this system of equations by elimination can be done as follows:\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "a + b + c = 100 (equation 1)<BR>\n" );
document.write( ".05a + .15b + .10c = 8.05 (equation 2)<BR>\n" );
document.write( ".10a + .08b + .04c = 6.84 (equation 3)\r<BR>\n" );
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document.write( "work with equations 1 and 2 to start.\r<BR>\n" );
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document.write( "multiply both sides of equation 1 by .05 to get:\r<BR>\n" );
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document.write( ".05a + .05b + .05c = 5 (equation 4)<BR>\n" );
document.write( ".05a + .15b + .10c = 8.05 (equation 2)\r<BR>\n" );
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document.write( "subtract equation 4 from equation 2 to get:\r<BR>\n" );
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document.write( ".10b + .05c = 3.05 (equation 5)\r<BR>\n" );
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document.write( "now work with equation 1 and equation 3.\r<BR>\n" );
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document.write( "multiply both sides of equation 1 by .10 to get:\r<BR>\n" );
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document.write( ".10a + .10b + .10c = 10 (equation 6)<BR>\n" );
document.write( ".10a + .08b + .04c = 6.84 (equation 3)\r<BR>\n" );
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document.write( "subtract equation 3 from equation 6 to get:\r<BR>\n" );
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document.write( ".02b + .06c = 3.16 (equation 7)\r<BR>\n" );
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document.write( "you now have 2 equations in 2 unknowns.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( ".10b + .05c = 3.05 (equation 5)<BR>\n" );
document.write( ".02b + .06c = 3.16 (equation 7)\r<BR>\n" );
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document.write( "multiply both sides of equation 7 by 5 to get:\r<BR>\n" );
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document.write( ".10b + .05c = 3.05 (equation 5)<BR>\n" );
document.write( ".10b + .30c + 15.8 (equation 8)\r<BR>\n" );
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document.write( "subtract equation 5 from equation 8 to get:\r<BR>\n" );
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document.write( ".25c = 12.75\r<BR>\n" );
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document.write( "solve for c to get:\r<BR>\n" );
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document.write( "c = 51\r<BR>\n" );
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document.write( "now that you have c = 51, you can go back to equation 5 or equation 8 and solve for b.\r<BR>\n" );
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document.write( "we'll use equation 5.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( ".10b + .05c = 3.05 (equation 5) becomes:<BR>\n" );
document.write( ".10b + .05(51) = 3.05\r<BR>\n" );
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document.write( "solve for b to get:\r<BR>\n" );
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document.write( ".10b = 3.05 - .05(51) = .5\r<BR>\n" );
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document.write( "solve for b to get b = 5\r<BR>\n" );
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document.write( "you now have c = 51 and b = 5\r<BR>\n" );
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document.write( "go back to any of the original equations and solve for a.\r<BR>\n" );
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document.write( "we'll use equation 1.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "a + b + c = 100 (equation 1) becomes:\r<BR>\n" );
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document.write( "a + 5 + 51 = 100 which becomes:\r<BR>\n" );
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document.write( "a + 56 = 100\r<BR>\n" );
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document.write( "solve for a to get a = 44.\r<BR>\n" );
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document.write( "you now have:\r<BR>\n" );
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document.write( "a = 44<BR>\n" );
document.write( "b = 5<BR>\n" );
document.write( "c = 51\r<BR>\n" );
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document.write( "go back to your original equations and solve them using those values to see that all the equations are true.\r<BR>\n" );
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document.write( "they are, so the solution is correct.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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