document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=249149>Question 249149</A>: <I><SPAN STYLE=\"word-break: break-all;\"> by weight one fertilizer is 20% potassium, 30% nitrogen and 50% phosphorus. a second fertilizer has percents of 10%, 20% and 70% respectively and a third fertilizer has percentage of 0%, 30% and 70% respectively. how much of each should be mixed to get 200 kg of fertilizers with percents of 12%, 25% and 63% respectively? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #181524 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=181524\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x = amount of fertilizer 1<BR>\n" );
document.write( "y = amount of fertilizer 2<BR>\n" );
document.write( "z = amount of fertilizer 3\r<BR>\n" );
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document.write( "it helps to make a table for this.\r<BR>\n" );
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document.write( "your table would look like this:\r<BR>\n" );
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document.write( "amount of fertilizer in kilograms\r\n" );
document.write( "fertilizer     total     potassium     nitrogen     phosphorus\r\n" );
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document.write( "1                x          .2x          .3x           .5x\r\n" );
document.write( "2                y          .1y          .2y           .7y\r\n" );
document.write( "3                z          ---          .3z           .7z \r\n" );
document.write( "4              200           24           50           126\r\n" );
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document.write( "the amounts of potassium, nitrogen, and phosphorous for fertilizer 4 are given to you by weight because you are told the proportions and you know what the total weight is.\r<BR>\n" );
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document.write( "potassium = .12 * 200 = 24<BR>\n" );
document.write( "nitrogen = .25 * 200 = 50<BR>\n" );
document.write( "phosphorus = .63 * 200 = 126\r<BR>\n" );
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document.write( "you wind up with 3 equations that need to be solved simultaneously.\r<BR>\n" );
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document.write( "note that these equations are from the vertical columns for potassium, nitrogen, and phosphorus in your original table above.\r<BR>\n" );
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document.write( "those equations are:\r<BR>\n" );
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document.write( "           .2x + .1y + .0z = 24  (equation 1)\r\n" );
document.write( "           .3x + .2y + .3z = 50  (equation 2)\r\n" );
document.write( "           .5x + .7y + .7z = 126 (equation 3)\r\n" );
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document.write( "once you solve these simultaneously, you will have the amounts of x,y,z that you need which are the amounts of fertilizer 1, fertilizer 2, fertilizer 3 that you need.\r<BR>\n" );
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document.write( "we will eliminate the decimals first so they won't bother us.\r<BR>\n" );
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document.write( "multiply all equations by 10 to get:\r<BR>\n" );
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document.write( "           2x + 1y + 0z = 240  (equation 1)\r\n" );
document.write( "           3x + 2y + 3z = 500  (equation 2)\r\n" );
document.write( "           5x + 7y + 7z = 1260 (equation 3)\r\n" );
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document.write( "first we will take equation 1 and 2 and eliminate the y variable.\r<BR>\n" );
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document.write( "we will do that by multiplying equation 1 by 2 and then subtract equation 2 from equation 1 to get equation 4.\r<BR>\n" );
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document.write( "           4x + 2y + 0z =  480  (equation 1 multiplied by 2)\r\n" );
document.write( " minus:    3x + 2y + 3z =  500  (equation 2)\r\n" );
document.write( " equals:   1x      - 3z =  -20  (equation 4)\r\n" );
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document.write( "next we will take equation 2 and 3 and eliminate the y variable.\r<BR>\n" );
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document.write( "we will do that by multiplying equation 2 by 7 and multiplying equation 3 by 2 and then subtracting equation 3 from equation 2.\r<BR>\n" );
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document.write( "           21x + 14y + 21z = 3500  (equation 2 multiplied by 7)\r\n" );
document.write( " minus:    10x + 14y + 14z = 2520  (equation 3 multiplied by 2)\r\n" );
document.write( " equals:   11x       +  7z = 980   (equation 5)\r\n" );
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document.write( "next we will take equation 4 and equation 5 and eliminate the x variable.\r<BR>\n" );
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document.write( "we will do that by multiplying equation 4 by 11 and then subtracting equation 5 from equation 4\r<BR>\n" );
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document.write( "                  11x - 33z =  -220   (equation 4 multiplied by 11)\r\n" );
document.write( "      minus:      11x +  7z =   980   (equation 5)\r\n" );
document.write( "      equals:         - 40z = -1200\r\n" );
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document.write( "we solve for z to get:\r<BR>\n" );
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document.write( "z = 30\r<BR>\n" );
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document.write( "we use this value of z and substitute in equation 5 to get:\r<BR>\n" );
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document.write( "11x +  7z =   980 becomes:<BR>\n" );
document.write( "11x + 7*(30) = 980 which becomes:<BR>\n" );
document.write( "11x + 210 = 980 <BR>\n" );
document.write( "subtract 210 from both sides to get:<BR>\n" );
document.write( "11x = 980-210 = 770<BR>\n" );
document.write( "divide both sides by 11 to get:<BR>\n" );
document.write( "x = 70\r<BR>\n" );
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document.write( "next we use the values of z = 30 and x = 70 and go to equation 3 to get:\r<BR>\n" );
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document.write( "5x + 7y + 7z = 1260 becomes:<BR>\n" );
document.write( "5*(70) + 7y + 7*30) = 1260 which becomes:<BR>\n" );
document.write( "350 + 7y + 210 = 1260 which becomes<BR>\n" );
document.write( "7y + 560 = 1260<BR>\n" );
document.write( "subtract 560 from both sides to get:<BR>\n" );
document.write( "7y = 1260 - 560 = 700<BR>\n" );
document.write( "divide both sides by 7 to get:<BR>\n" );
document.write( "y = 100\r<BR>\n" );
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document.write( "we have values for x,y,z.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "x = 70<BR>\n" );
document.write( "y = 100<BR>\n" );
document.write( "z = 30\r<BR>\n" );
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document.write( "we need to test these values to see if they are good.\r<BR>\n" );
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document.write( "our original equations are:\r<BR>\n" );
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document.write( "           .2x + .1y + .0z = 24  (equation 1)\r\n" );
document.write( "           .3x + .2y + .3z = 50  (equation 2)\r\n" );
document.write( "           .5x + .7y + .7z = 126 (equation 3)\r\n" );
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document.write( "first we check to see if x + y + z = 200.\r<BR>\n" );
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document.write( "it does.\r<BR>\n" );
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document.write( "that's good because the total of fertilizer 1, 2, 3 should equal 200 and it does.\r<BR>\n" );
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document.write( "next we substitute in all 3 equations to get:\r<BR>\n" );
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document.write( "           .2x + .1y + .0z = 24  (equation 1)\r\n" );
document.write( "           .3x + .2y + .3z = 50  (equation 2)\r\n" );
document.write( "           .5x + .7y + .7z = 126 (equation 3)\r\n" );
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document.write( "becomes:<BR>\n" );
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document.write( "           .2*70 + .1*100 + .0*30 = 24  (equation 1)\r\n" );
document.write( "           .3*70 + .2*100 + .3*30 = 50  (equation 2)\r\n" );
document.write( "           .5*70 + .7*100 + .7*30 = 126 (equation 3)\r\n" );
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document.write( "we simplify to get:<BR>\n" );
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document.write( "           14 +  10 +  0 =  24  (equation 1)\r\n" );
document.write( "           21 +  20 +  9 =  50  (equation 2)\r\n" );
document.write( "           35 +  70 + 21 = 126  (equation 3)\r\n" );
document.write( "           -------------------\r\n" );
document.write( "  total:   70 + 100 + 30 = 200  (sum of equations 1,2,3)\r\n" );
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document.write( "Looks like we have the answer.\r<BR>\n" );
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document.write( "he needs:\r<BR>\n" );
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document.write( "70 pounds of fertilizer 1<BR>\n" );
document.write( "100 pounds of fertilizer 2<BR>\n" );
document.write( "30 pounds of fertilizer 3\r<BR>\n" );
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document.write( "he will get:\r<BR>\n" );
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document.write( "200 pounds of fertilizer 4 that contains:\r<BR>\n" );
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document.write( "24 pounds of potassium <BR>\n" );
document.write( "50 pounds of nitrogen<BR>\n" );
document.write( "126 pounds of phosphorus\r<BR>\n" );
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document.write( " </SPAN>\n" );
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