document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=235629>Question 235629</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Chemistry. A chemist has two solutions: one containing 40% alcohol and another containing 70% alcohol. How much of each should be used to obtain 80 liters of a 49% solution?\r<BR>\n" );
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document.write( "Im trying to find a way to solve this using matrices. I'm really unsure about how to go around solving this problem. Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #173510 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=173510\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The solution that you want will equal 80 liters.\r<BR>\n" );
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document.write( "The amount of alcohol that you want in this solution would therefore be .49*80 = 39.2 liters of alcohol.\r<BR>\n" );
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document.write( "Let x = the number of liters in the first solution that you have.\r<BR>\n" );
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document.write( "Let y = the number of liters in the second solution that you have.\r<BR>\n" );
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document.write( "The amount of alcohol in the first solution that you have is therefore .4*x\r<BR>\n" );
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document.write( "The amount of alcohol in the second solution that you have is therefore .7*y\r<BR>\n" );
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document.write( "You will mix these together to get 39.2 liters of alcohol which represents .49 of your new solution.\r<BR>\n" );
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document.write( "We have an equation that says:\r<BR>\n" );
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document.write( ".4*x + .7*y = 39.2\r<BR>\n" );
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document.write( "We have another equation that says that:\r<BR>\n" );
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document.write( "x + y = 80\r<BR>\n" );
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document.write( "The first equation tells us how much alcohol is in the new solution.\r<BR>\n" );
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document.write( "The second equation tells us how much total solution is in the new solution.\r<BR>\n" );
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document.write( "This has to be since the 80 liters are coming from x and y only.\r<BR>\n" );
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document.write( "These equations need to be solved simultaneously.\r<BR>\n" );
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document.write( "The equations are:\r<BR>\n" );
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document.write( ".4*x + .7*y = 39.2\r<BR>\n" );
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document.write( "x + y = 80\r<BR>\n" );
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document.write( "We can solve by substitution or by elimination.\r<BR>\n" );
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document.write( "You can also solve using matrix algebra if you wish.\r<BR>\n" );
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document.write( "The equations should be in the form you can use to do that.\r<BR>\n" );
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document.write( "I'll use substitution.\r<BR>\n" );
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document.write( "Since x + y = 80, I'll solve for y to get:\r<BR>\n" );
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document.write( "y = 80 - x\r<BR>\n" );
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document.write( "I'll substitute in the equation:\r<BR>\n" );
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document.write( ".4*x + .7*y = 39.2 to get:\r<BR>\n" );
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document.write( ".4*x + .7*(80-x) = 39.2\r<BR>\n" );
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document.write( "Expand parentheses to get:\r<BR>\n" );
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document.write( ".4*x + .7*80 - .7*x = 39.2\r<BR>\n" );
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document.write( "Simplify to get:\r<BR>\n" );
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document.write( ".4*x + 56 - .7*x = 39.2\r<BR>\n" );
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document.write( "Combine like terms to get:\r<BR>\n" );
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document.write( "-.3*x + 56 = 39.2\r<BR>\n" );
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document.write( "Subtract 56 from both sides of the equation to get:\r<BR>\n" );
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document.write( "-.3*x = 39.2 - 56 = -16.8\r<BR>\n" );
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document.write( "Divide both sides by -.3 to get:\r<BR>\n" );
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document.write( "x = 56\r<BR>\n" );
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document.write( "Since x = 56 and x + y = 80, this means that:\r<BR>\n" );
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document.write( "y + 56 = 80 which means that y = 24\r<BR>\n" );
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document.write( "We have:\r<BR>\n" );
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document.write( "x = 56<BR>\n" );
document.write( "y = 24\r<BR>\n" );
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document.write( "Plug these values into the original equations and see if they work out.\r<BR>\n" );
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document.write( "The original equations are:\r<BR>\n" );
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document.write( ".4*x + .7*y = 39.2\r<BR>\n" );
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document.write( "x + y = 80\r<BR>\n" );
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document.write( "replace x with 56 and y with 24 to get:\r<BR>\n" );
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document.write( ".4*56 + .7*24 = 39.2\r<BR>\n" );
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document.write( "56 + 24 = 80\r<BR>\n" );
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document.write( "The second equation becomes 80 = 80 so that one is good.\r<BR>\n" );
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document.write( "The first equation becomes:\r<BR>\n" );
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document.write( "22.4 + 16.8 = 39.2 which becomes:\r<BR>\n" );
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document.write( "39.2 = 39.2 so the first part is good two.\r<BR>\n" );
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document.write( "The answer to your probem is:\r<BR>\n" );
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document.write( "You need 56 liters of the .4 solution and 24 liters of the .7 solution to make 80 liters of a .49 solution.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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