document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=367546>Question 367546</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 10 liters of a 30% acid solution is obtained by mixing 20% solution with a 50% solution.  \r<BR>\n" );
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document.write( "a) write a system of equations in which one equation represents the amount of final mixture required and the other represents the amount of the acid in the final mixture.  Let x and y represent the amounts of 20% and 50% solutions, respectively.\r<BR>\n" );
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document.write( "b) use a graphing utility to graph the two equations in part a.  As the amount of the 20% solution increases, how does the amount of the 50% solution change?\r<BR>\n" );
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document.write( "c) How much of each solution is required to obtain the specified concentration in the final mixture?<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 #262077 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=262077\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x is the amount of liters of the 20% solution.<BR>\n" );
document.write( "y is the amount of liters of the 50% solution.\r<BR>\n" );
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document.write( "your 2 equations are:\r<BR>\n" );
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document.write( "x + y = 10 (number of liters in the solution)\r<BR>\n" );
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document.write( ".2x + .5y = .3*10 (amount of acid in the solution).\r<BR>\n" );
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document.write( "you solve these simultaneously as follows:\r<BR>\n" );
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document.write( "from the first equation, solve for y to get:\r<BR>\n" );
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document.write( "y = 10 - x\r<BR>\n" );
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document.write( "substitute for y in the second equation to get:\r<BR>\n" );
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document.write( ".2*x + .5*(10-x) = .3*10\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( ".2*x + 5 - .5*x = 3\r<BR>\n" );
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document.write( "combine like terms to get -.3*x + 5 = 3\r<BR>\n" );
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document.write( "subtract 5 from both sides of the first equation to get:\r<BR>\n" );
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document.write( "-.3*x = -2\r<BR>\n" );
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document.write( "divide both sides of the equation by -.3 to get:\r<BR>\n" );
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document.write( "x = 6.6666666667 (equivalent to 6 and 2/3)\r<BR>\n" );
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document.write( "since x + y = 10, this means that y = 3.333333333 (equivalent to 3 and 1/3)\r<BR>\n" );
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document.write( "substitute in your original equations to get:\r<BR>\n" );
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document.write( "x + y = 6.6666666667 + 3.3333333333 = 10\r<BR>\n" );
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document.write( ".2*x + .5*y = .2 * (6.6666666667) + .5 * (3.3333333333) = 3\r<BR>\n" );
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document.write( "your solution to the problem is that you need 6 and 2/3 liters of the 20% solution and 3 and 1/3 liters of the 50% solution to get a solution of 10 liters that is 30% acid.\r<BR>\n" );
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document.write( "to graph this problem, you need to do the following:\r<BR>\n" );
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document.write( "Let x = the number of liters of the 20% solution.\r<BR>\n" );
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document.write( "Let y = .2*x + .5*(10-x) which equals the amount of acid in 10 liters of the final solution.\r<BR>\n" );
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document.write( "the graph will look like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-1%2C11%2C-1%2C7%2C.2%2Ax+%2B+.5%2A%2810-x%29%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-1%2C11%2C-1%2C7%2C.2%2Ax+%2B+.5%2A%2810-x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The domain of your graph is the value of x from 0 to 10.\r<BR>\n" );
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document.write( "The range of your graph is the value of y from 5 to 2.\r<BR>\n" );
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document.write( "you need to look only at the values of x from 0 to 10.\r<BR>\n" );
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document.write( "It's clear that when x = 0, y = 5.\r<BR>\n" );
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document.write( "That's because your mix is .2*0 + .5*10\r<BR>\n" );
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document.write( "It's also clear that when x = 10, y = 2\r<BR>\n" );
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document.write( "That's because your mix is .2*10 + .5*0\r<BR>\n" );
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document.write( "It is a little less clear that when the value of x = 6.6666666667, the value of y is equal to 3.\r<BR>\n" );
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document.write( "It's a little less clear because you can't really narrow the graph down to the value of x = 6.66666667.\r<BR>\n" );
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document.write( "You can, however, determine the value of y from the equation for y.\r<BR>\n" );
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document.write( "The equation is:\r<BR>\n" );
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document.write( "y = .2*x + .5*(10-x)\r<BR>\n" );
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document.write( "That becomes:\r<BR>\n" );
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document.write( "y = .2*6.666666667 + .5 * (10-6.666666667)\r<BR>\n" );
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document.write( "The result of that calculation is that y = 3.\r<BR>\n" );
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document.write( "The graph above showed you what happens to the amount of acid in the 10 liters of total solution as the number of liters of the 20% solution increases.\r<BR>\n" );
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document.write( "The following graph shows you what happens to the number of liters of 50% solution as the number of liters of the 20% solution increases.\r<BR>\n" );
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document.write( "In this graph, the value of x is the number of liters of 20% solution and the value of y is the number of liters of 50% solution.\r<BR>\n" );
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document.write( "The formula for y is y = 10 - x.\r<BR>\n" );
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document.write( "as x increases, the value of y decreases.\r<BR>\n" );
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document.write( "That graph looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-1%2C11%2C-1%2C11%2C10-x%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-1%2C11%2C-1%2C11%2C10-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can see that when the amount of the 20% solution is increased, the amount of the 50% solution decreases.\r<BR>\n" );
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document.write( "x is the amount of 20% solution.<BR>\n" );
document.write( "y is the amount of 50% solution.\r<BR>\n" );
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