document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=351114>Question 351114</A>: <I><SPAN STYLE=\"word-break: break-all;\">  PLS. HELP ME SOLVE THIS PROBLEM. TNX.\r<BR>\n" );
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document.write( "1. A man divides P100,000 among his three investments, at 4%, 5%, and 6% per annum, respectively. The total income is P4,900 per year. His income from the 5% and 6% investments exceeds the income at 4% by P 2,500. Find how much is invested at each rate.<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 #250959 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=250959\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Total investment is 100,000<BR>\n" );
document.write( "x amount of investment is invested at 4% per year.<BR>\n" );
document.write( "y amount of investment is invested at 5% per year.<BR>\n" );
document.write( "z amount of investment is invested at 6% per year.\r<BR>\n" );
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document.write( "income from y and z exceeds income from x by 2,500.\r<BR>\n" );
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document.write( "first equation states that the total investment is composed of investment x, y, and z, and totals 100,000.\r<BR>\n" );
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document.write( "x + y + z = 100,000 (first equation)\r<BR>\n" );
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document.write( "second equation states that income from all of these investments equals 4,900 per year.\r<BR>\n" );
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document.write( ".04*x + .05*y + .06*z = 4,900 (second equation)\r<BR>\n" );
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document.write( "you have 2 equations in 3 unknowns.\r<BR>\n" );
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document.write( "we'll try to reduce the number of unknowns by establishing some equivalencies.\r<BR>\n" );
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document.write( "you are also given that the income from investment y and investment z exceeds the income from investment x by 2,500.\r<BR>\n" );
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document.write( "the equation to express that relationship is shown below:\r<BR>\n" );
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document.write( ".05*y + .06*z = .04*x + 2,500\r<BR>\n" );
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document.write( "we can use this equation to reduce the number of unknowns in the first 2 equations.\r<BR>\n" );
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document.write( "we'll solve for .04*x to get:\r<BR>\n" );
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document.write( ".04*x = .05*y + .06*z - 2,500\r<BR>\n" );
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document.write( "we'll solve for x to get:\r<BR>\n" );
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document.write( "x = (.05*y + .06*z - 2,500) / .04\r<BR>\n" );
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document.write( "we now have values for x and .04*x in terms of y and z.\r<BR>\n" );
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document.write( "we'll take the first equation and the second equation and substitute for x and .04*x in them as follows:\r<BR>\n" );
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document.write( "the equations are:\r<BR>\n" );
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document.write( "x + y + z = 100,000 (first equation)<BR>\n" );
document.write( ".04*x + .05*y + .06*z = 4,900 (second equation)\r<BR>\n" );
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document.write( "substituting for x and .04*x in these equations gets us:\r<BR>\n" );
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document.write( "((.05*y + .06*z - 2,500) / .04) + y + z = 100,000<BR>\n" );
document.write( "and:<BR>\n" );
document.write( "(.05*y + .06*z - 2,500) + .05*y + .06*z = 4,900\r<BR>\n" );
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document.write( "if we solve these 2 equations in 2 unknowns simultaneously, we should be able to get our answer.\r<BR>\n" );
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document.write( "first step is to simplify these equations.\r<BR>\n" );
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document.write( "modified first equation is:\r<BR>\n" );
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document.write( "((.05*y + .06*z - 2,500) / .04) + y + z = 100,000\r<BR>\n" );
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document.write( "modified first equation simplifies to:\r<BR>\n" );
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document.write( "1.25*y + 1.5*z - 62,500 + y + z = 100,000\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "2.25*y + 2.5*z - 62,500 = 100,000\r<BR>\n" );
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document.write( "add 62,500 to both sides of the equation to get:\r<BR>\n" );
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document.write( "2.25*y + 2.5*z = 162,500 (modified first equation)\r<BR>\n" );
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document.write( "modified second equation is:\r<BR>\n" );
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document.write( "(.05*y + .06*z - 2,500) + .05*y + .06*z = 4,900\r<BR>\n" );
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document.write( "modified second equation simplifies to:\r<BR>\n" );
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document.write( ".05*y + .06*z - 2,500 + .05*y + .06*z = 4,900\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( ".10*y + .12*z - 2,500 = 4,900\r<BR>\n" );
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document.write( "add 2,500 to both sides of the equation to get:\r<BR>\n" );
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document.write( ".10*y + .12*z = 7,400 (modified second equation).\r<BR>\n" );
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document.write( "your 2 equations that now need to be solved simultaneously are:\r<BR>\n" );
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document.write( "2.25*y + 2.5*z = 162,500 (modified first equation)<BR>\n" );
document.write( "and:<BR>\n" );
document.write( ".10*y + .12*z = 7,400 (modified second equation).\r<BR>\n" );
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document.write( "Multiply both sides of the second equation by 22.5 to get:\r<BR>\n" );
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document.write( "2.25*y + 2.5*z = 162,500 (modified first equation)<BR>\n" );
document.write( "and:<BR>\n" );
document.write( "2.25*y + 2.7*z = 166,500 (modified second equation).\r<BR>\n" );
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document.write( "Subtract the first equation from the second equation to get:\r<BR>\n" );
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document.write( ".2*z = 4,000\r<BR>\n" );
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document.write( "Divide both sides of this equation by .2 to get:\r<BR>\n" );
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document.write( "z = 4,000 / .2 = 20,000\r<BR>\n" );
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document.write( "use this value of z to solve for y in either one of the modified first equation or modified second equation (the original modified before we started multiplying them out.\r<BR>\n" );
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document.write( "those 2 original modified equations are:\r<BR>\n" );
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document.write( "2.25*y + 2.5*z = 162,500 (modified first equation)<BR>\n" );
document.write( "and:<BR>\n" );
document.write( ".10*y + .12*z = 7,400 (modified second equation).\r<BR>\n" );
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document.write( "In the first of these equation, let z = 20,000 to get:\r<BR>\n" );
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document.write( "2.25*y + 2.5*(20,000) = 162,500\r<BR>\n" );
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document.write( "Simplify to get:\r<BR>\n" );
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document.write( "2.25*y + 50,000 = 162,500\r<BR>\n" );
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document.write( "Subtract 50,000 from both sides of this equation to get:\r<BR>\n" );
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document.write( "2.25*y = 162,500 - 50,000\r<BR>\n" );
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document.write( "Simplify further to get:\r<BR>\n" );
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document.write( "2.25*y = 112,500\r<BR>\n" );
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document.write( "Divide both sides of this equation by 2.25 to get:\r<BR>\n" );
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document.write( "y = 50,000\r<BR>\n" );
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document.write( "If you solved for y in the second equation, you would have gotten the same answer, as you should.   I checked it out and it's good.\r<BR>\n" );
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document.write( "You now have 2 values.\r<BR>\n" );
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document.write( "y = 50,000, and z = 20,000\r<BR>\n" );
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document.write( "Since x + y + z = 100,000, that must mean that x = 30,000\r<BR>\n" );
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document.write( "Your answer should be:\r<BR>\n" );
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document.write( "x = 30,000<BR>\n" );
document.write( "y = 50,000<BR>\n" );
document.write( "z = 20,000\r<BR>\n" );
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document.write( "It remains only to check your answers to confirm that they are good.\r<BR>\n" );
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document.write( "Your original first equation is:\r<BR>\n" );
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document.write( "x + y + z = 100,000\r<BR>\n" );
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document.write( "That's clearly true.\r<BR>\n" );
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document.write( "Your original second equation is:\r<BR>\n" );
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document.write( ".04*x + .05*y + .06*z = 4,900\r<BR>\n" );
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document.write( "Substituting for x,y,z that we solved for, this becomes:\r<BR>\n" );
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document.write( ".04*30,000 + .05*50,000 + .06*20,000 = 4,900\r<BR>\n" );
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document.write( "This simplifies to:\r<BR>\n" );
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document.write( "1,200 + 2,500 + 1,200 = 4,900\r<BR>\n" );
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document.write( "This simplifies further to:\r<BR>\n" );
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document.write( "4,900 = 4,900 which is also true.\r<BR>\n" );
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document.write( "The third equation that we used to reduce the number of unknowns to 2 was:\r<BR>\n" );
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document.write( ".05*y + .06*z = .04*x + 2,500\r<BR>\n" );
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document.write( "Substituting for x, y, and z in this equation, we get:\r<BR>\n" );
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document.write( ".05*50,000 + .06*20,000 = .04*30,000 + 2,500\r<BR>\n" );
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document.write( "This equation simplifies to:\r<BR>\n" );
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document.write( "2,500 + 1,200 = 1,200 + 2,500\r<BR>\n" );
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document.write( "This equation simplifies to:\r<BR>\n" );
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document.write( "3,700 = 3,700 which is also true.\r<BR>\n" );
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document.write( "Our answers for x,y,z are good.\r<BR>\n" );
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document.write( "x = investment at 4% = 30,000<BR>\n" );
document.write( "y = investment at 5% = 50,000<BR>\n" );
document.write( "z = investment at 6% = 20,000\r<BR>\n" );
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