document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=920759>Question 920759</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation: Sue invested $10,000 in three parts. With one part she bought mutual funds that offered a return of 4% per year. The second part which amounted to twice the first was used to buy government bonds paying 4.5% per year. She put the rest into an account paying 2.5% annual interest. During the first year the total interest was $415. How much did she invest at each rate?\r<BR>\n" );
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document.write( "I set up a table, but am not sure how to put the numbers together into an equation that I can work through. I have no idea how to set this up.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #558573 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=558573\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> sue invested a total of $10,000\r<BR>\n" );
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document.write( "let x + y + z equal the amount of money invested in the three parts.\r<BR>\n" );
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document.write( "x is for the first part<BR>\n" );
document.write( "y is for the second part<BR>\n" );
document.write( "z is for the third part\r<BR>\n" );
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document.write( "with the first part she bought mutual funds that returned 4% per year.<BR>\n" );
document.write( "that would be x and the return would be equal to .04 * x.\r<BR>\n" );
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document.write( "with the second part she bought governmen<BR>\n" );
document.write( "t bonds paying 4.5% per year.<BR>\n" );
document.write( "that would be y and the return would be equal to .045 * y.\r<BR>\n" );
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document.write( "you are also given that the second part was twice the size of the first part.<BR>\n" );
document.write( "that would make y = 2x.\r<BR>\n" );
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document.write( "hold on to that thought because you will use it later to reduce the number of variables involved.\r<BR>\n" );
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document.write( "with the third part she placed it into an account paying 2.5% annual interest.<BR>\n" );
document.write( "that would be z and the return would be equal to .025 * z.\r<BR>\n" );
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document.write( "during the first year she made $415 in interest.\r<BR>\n" );
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document.write( "you have 2 equations to work with and those equations need to be solved simultaneously.\r<BR>\n" );
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document.write( "the first equation is:\r<BR>\n" );
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document.write( "x + y + z = $10,000\r<BR>\n" );
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document.write( "this equation gives you the total amount invested by each part.\r<BR>\n" );
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document.write( "the second equation is:\r<BR>\n" );
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document.write( ".04 * x + .045 * y + .025 * z = $415.\r<BR>\n" );
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document.write( "this equation gives you the total interest earned by each part.\r<BR>\n" );
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document.write( "you have 2 equations in 3 variables which will not allow you to solve for a specific solution.\r<BR>\n" );
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document.write( "in order to solve for a specific solution, the number of equations needs to be less than or equal to the number of variables.\r<BR>\n" );
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document.write( "the preferred situation is that the number of equations is equal to the number of variables.\r<BR>\n" );
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document.write( "the equations that you want to solve simultaneously are:\r<BR>\n" );
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document.write( "x + y + z = $10,000<BR>\n" );
document.write( ".04 * x + .045 * y + .025 * z = $415\r<BR>\n" );
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document.write( "fortunately, you can eliminate one of the variables, because you are told that the second part is twice the size of the first part which leads to y = 2x.\r<BR>\n" );
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document.write( "this means you can replace y with 2x in x + y + z = $10,000 to get x + 2x + z = $10,000 which becomes 3x + z = $10,000.\r<BR>\n" );
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document.write( "this also means you can replace y with 2x in .04x + .045y + .025z = $415 to get .04x + .045*2x + .025*z = $415k which becomes .04x + .09x + .025z = $415 which becomes .13x + .025z = $415.\r<BR>\n" );
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document.write( "your 2 equations now only have 2 variables that need to be solved and you can solve for a specific solution.\r<BR>\n" );
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document.write( "those equations are:\r<BR>\n" );
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document.write( "3x + z = $10,000<BR>\n" );
document.write( ".13x + .025z = 415\r<BR>\n" );
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document.write( "in the first equation, solve for z to get:\r<BR>\n" );
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document.write( "z = $10,000 - 3x\r<BR>\n" );
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document.write( "in the second equation, replace z with $10,000 - 3x to get:\r<BR>\n" );
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document.write( ".13x + .025 * ($10,000 - 3x) = $415.\r<BR>\n" );
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document.write( "solve for x as follows:\r<BR>\n" );
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document.write( "simplify to get:<BR>\n" );
document.write( ".13x + .025*10,000 - .025*3x = 415<BR>\n" );
document.write( "simplify further to get:<BR>\n" );
document.write( ".13x + 250 - .075x = 415<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( ".055x + 250 = 415<BR>\n" );
document.write( "subtract 250 from both sides of this equation to get:<BR>\n" );
document.write( ".055x = 165<BR>\n" );
document.write( "divide both sides of this equation by .055 to get:<BR>\n" );
document.write( "x = 3000.\r<BR>\n" );
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document.write( "since y = 2x, then y = 6000.\r<BR>\n" );
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document.write( "since the total is 10,000, then z = 1000.\r<BR>\n" );
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document.write( "you have:\r<BR>\n" );
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document.write( "x = 3,000<BR>\n" );
document.write( "y = 6,000<BR>\n" );
document.write( "z = 1,000\r<BR>\n" );
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document.write( "x + y + z = $10,000 so that part's good.\r<BR>\n" );
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document.write( ".04x + .045y + .025z = 120 + 270 + 25 = $415 so that part's good.\r<BR>\n" );
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document.write( "solution looks good.\r<BR>\n" );
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document.write( "solution is:\r<BR>\n" );
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document.write( "she invested $3,000 at 4% and $6,000 at 4.5% and $1,000 at 2.5%.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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