document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=127521>Question 127521</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/053445318X/algebrahomeworkh>College Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I am having a hard time setting up two equations for this word problem:  A shopkeeper has two types of coffee beans.  The first type sells for $5.20/lb, the second for $5.80.  How many pounds of the first type must be mixed with 5 lb of the second to produce a blend selling for $5.35/lb?\r<BR>\n" );
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document.write( "I have attempted setting up on equation $5.20(a) + $5.80(y) = $5.35 (1), and several other ways, and I just cannot seem to set this up algebraically.  Can you please help me set up the equations so that I can attempt to solve this problem? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #93457 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=93457\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The problem requires two equations to solve.<BR>\n" );
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document.write( "The first equation will involve the total amount of mixed coffee. The total amount of mixed coffee<BR>\n" );
document.write( "(call it T for total) will consist of X, the unknown number of lbs of the $5.20/lb coffee, and<BR>\n" );
document.write( "5 lbs of the $5.80/lb coffee. So we can say that T, the weight of the blend is the sum of X<BR>\n" );
document.write( "and 5 lbs and in equation form this is:<BR>\n" );
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document.write( "T = X + 5<BR>\n" );
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document.write( "Then we can determine the cost. We know that the cost of the blend is to be $5.35 per lb. Since<BR>\n" );
document.write( "we have T lbs of the blend, the value of the blend is $5.35 per lb times T, the number of<BR>\n" );
document.write( "lbs of the blend ... or 5.35 times T ... or just 5.35T.<BR>\n" );
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document.write( "The cost of the 5 lbs of $5.80/lb coffee is just the product of $5.80 times 5 lbs or<BR>\n" );
document.write( "5.80 times 5 which multiplies out to $29.00. <BR>\n" );
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document.write( "And finally the cost of the X lbs of coffee at $5.20/lb is the product of these two or 5.20X.<BR>\n" );
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document.write( "Then the dollar value of the blend must equal the dollar value of the two types of coffee in the blend.<BR>\n" );
document.write( "In equation form this is:<BR>\n" );
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document.write( "5.35T = 5.20X + 29.00<BR>\n" );
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document.write( "Then we can go back to the first equation T = X + 5. Since we know that T is equal to<BR>\n" );
document.write( "X + 5 we can substitute X + 5 for T in the cost equation and get:<BR>\n" );
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document.write( "5.35(X + 5) = 5.20X + 29<BR>\n" );
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document.write( "Multiply out the left side by multiplying 5.35 times each of the terms in the parentheses to<BR>\n" );
document.write( "get:<BR>\n" );
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document.write( "5.35X + 5.35*5 = 5.20X + 29.00<BR>\n" );
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document.write( "The product of 5 and 5.35 is 26.75. So the equation becomes:<BR>\n" );
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document.write( "5.35X + 26.75 = 5.20X + 29.00<BR>\n" );
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document.write( "Get rid of the 26.75 on the left side by subtracting 26.75 from both sides. This subtraction<BR>\n" );
document.write( "changes the equation to:<BR>\n" );
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document.write( "5.35X = 5.20X + 2.25<BR>\n" );
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document.write( "Get rid of the 5.20X on the right side by subtracting 5.20X from both sides and the equation<BR>\n" );
document.write( "reduces to:<BR>\n" );
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document.write( "0.15X = 2.25<BR>\n" );
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document.write( "Finally solve for X by dividing both sides by 0.15 to get:<BR>\n" );
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document.write( "X = 2.25/0.15 = 15<BR>\n" );
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document.write( "This tells you that if you mix 15 lbs of coffee at $5.20 per lb with 5 lbs of coffee at<BR>\n" );
document.write( "$5.80 per lb you should get a blend that retails for $5.35 per lb.<BR>\n" );
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document.write( "Let's check that. We mix in 15 lbs of cheap coffee with 5 lbs of expensive coffee and<BR>\n" );
document.write( "so we get 20 lbs of mixed coffee. The 20 lbs of mixture is supposed to be worth $5.35 per lb<BR>\n" );
document.write( "and since we have 20 pounds of that blend the value of the coffee will be $5.35 times 20<BR>\n" );
document.write( "which, when you multiply it out is $107.00.<BR>\n" );
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document.write( "How much was the coffee that we mixed worth? First we took 15 lbs of coffee that was worth<BR>\n" );
document.write( "$5.20 per lb. So the worth of this coffee was 15 times $5.20 which multiplies out to <BR>\n" );
document.write( "$78.00. To that we added 5 lbs of coffee that was worth $5.80 per lb. That multiplies<BR>\n" );
document.write( "out to be $5.80 times 5 or $29.00. Therefore, the two coffees that we mixed were worth a total<BR>\n" );
document.write( "of $78.00 + $29.00 = $107.00 which is the same amount of worth that we figured for the blended<BR>\n" );
document.write( "mix. So our answer checks. The unknown number of lbs of the coffee at $5.20 per lb is 15 lbs.<BR>\n" );
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document.write( "Hope this helps you to understand how to develop the formulas and how to think your way through<BR>\n" );
document.write( "the problem. The two equations you needed were an equation for total weight of the mix as<BR>\n" );
document.write( "being the sum of the individual weights of the two coffees. And an equation that tells you<BR>\n" );
document.write( "the weight of the mixed coffee at its cost per pound equals the sum of the number of <BR>\n" );
document.write( "pounds of each type of coffee times the associated cost of each type.<BR>\n" );
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document.write( " </SPAN>\n" );
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