document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=855172>Question 855172</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Divide 8 into two parts such that the sum of their squares is 34.\r<BR>\n" );
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document.write( "2. $9000 were divided equally among a certain number of persons. Had there been 20 persons more, each would have got $160 less. Find the original number of persons. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #515127 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=515127\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = one of the parts.<BR>\n" );
document.write( "let y = the other of the parts.<BR>\n" );
document.write( "you get:<BR>\n" );
document.write( "x*y = 8 ***** first equation<BR>\n" );
document.write( "x^2 + y^2 = 34 ***** second equation<BR>\n" );
document.write( "solve for x in the first equation to get:<BR>\n" );
document.write( "x = 8/y<BR>\n" );
document.write( "substitute for x in the second equation to get:<BR>\n" );
document.write( "(8/y)^2 + y^2 = 34<BR>\n" );
document.write( "simplify this equation to get:<BR>\n" );
document.write( "64/y^2 + y^2 = 34<BR>\n" );
document.write( "multiply both sides of this equation by y^2 to get:<BR>\n" );
document.write( "64 + y^4 = 34y^2<BR>\n" );
document.write( "reorder the terms to get:<BR>\n" );
document.write( "y^4 + 64 = 34y^2<BR>\n" );
document.write( "subtract 34y^2 from both sides of this equation to get:<BR>\n" );
document.write( "y^4 - 34y^2 + 64 = 0<BR>\n" );
document.write( "factor this equation to get:<BR>\n" );
document.write( "(y^2 - 32) * (y^2 - 2) = 0<BR>\n" );
document.write( "solve for y^2 in each of the factors to get:<BR>\n" );
document.write( "y^2 = 32<BR>\n" );
document.write( "y^2 = 2<BR>\n" );
document.write( "solve for y in each of these equations to get:<BR>\n" );
document.write( "y = + sqrt(32)<BR>\n" );
document.write( "y = - sqrt(32)<BR>\n" );
document.write( "y = + sqrt(2)<BR>\n" );
document.write( "y = - sqrt(2)<BR>\n" );
document.write( "for each of these values of y, solve for x using the first equation.<BR>\n" );
document.write( "first equation is:<BR>\n" );
document.write( "x * y = 8<BR>\n" );
document.write( "solve for x in this equation to get:<BR>\n" );
document.write( "x = 8/y<BR>\n" );
document.write( "use the values of y you just solved for to get the (x,y) pairs that will be the possible solutions to this problem.<BR>\n" );
document.write( "you get:<BR>\n" );
document.write( "when y = + sqrt(32), x = + 8/sqrt(32)<BR>\n" );
document.write( "when y = - sqrt(32), x = - 8/sqrt(32)<BR>\n" );
document.write( "when y = + sqrt(2), x = + 8/sqrt(2)<BR>\n" );
document.write( "when y = - sqrt(2), x = - 8/sqrt(2)\r<BR>\n" );
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document.write( "those are your possible solutions.\r<BR>\n" );
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document.write( "you can confirm by multiplying x * y to make sure you get 8.<BR>\n" );
document.write( "you can also confirm by adding x^2 + y^2 to make sure you get 34.<BR>\n" );
document.write( "i'll do one of the possible solutions to show you.<BR>\n" );
document.write( "you can do the rest.<BR>\n" );
document.write( "when x = 8/sqrt(32) and y = sqrt(32), you get:<BR>\n" );
document.write( "x * y = sqrt(32) * 8/sqrt(32) which is equal to 8.<BR>\n" );
document.write( "x^2 + y^2 = (8/sqrt(32))^2 + sqrt(32)^2 which is equal to:<BR>\n" );
document.write( "8^2 / sqrt(32)^2 + sqrt(32)^2 which is equal to:<BR>\n" );
document.write( "64 / 32 + 32 which is equal to:<BR>\n" );
document.write( "2 + 32 which is equal to 34.\r<BR>\n" );
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document.write( "you can also graph your original equations to find the intesection points.<BR>\n" );
document.write( "those intersection points will agree with the solution just shown above.<BR>\n" );
document.write( "that graph looks like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-10%2C10%2C-10%2C10%2C8%2Fx%2Csqrt%2834-x%5E2%29%2C-sqrt%2834-x%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "sqrt(32) is equal to approximately 5.7<BR>\n" );
document.write( "sqrt(2) is equal to approximately 1.4<BR>\n" );
document.write( "8/sqrt(32) is equal to approximately 1.4<BR>\n" );
document.write( "8/sqrt(2) is equal to approximately 5.7\r<BR>\n" );
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document.write( "note that sqrt(32) is the same as 8/sqrt(2) and sqrt(2) is the same as 8/sqrt(32).<BR>\n" );
document.write( "this could throw you off but that's perfectly legitimate in this case.<BR>\n" );
document.write( "i'll show you one of them and you can prove the other one to yourself.\r<BR>\n" );
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document.write( "i'll show you why sqrt(32) is the same as 8 / sqrt(2)<BR>\n" );
document.write( "start with sqrt(32) = 8/sqrt(2)<BR>\n" );
document.write( "square both sides of this equation to get:<BR>\n" );
document.write( "32 = 64 / 2<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "32 = 32<BR>\n" );
document.write( "this confirms they are equivalent.<BR>\n" );
document.write( "another way to confirm it is:<BR>\n" );
document.write( "start with sqrt(32) = 8/sqrt(2)<BR>\n" );
document.write( "multiply both sides of this equation by sqrt(2) to get:<BR>\n" );
document.write( "sqrt(32) * sqrt(2) = 8<BR>\n" );
document.write( "since sqrt(a) * sqrt(b) = sqrt(a*b), this equation becomes:<BR>\n" );
document.write( "sqrt(32*2) = 8<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "sqrt(64) = 8<BR>\n" );
document.write( "simplify further to get:<BR>\n" );
document.write( "8 = 8<BR>\n" );
document.write( "the equality is confirmed again, only using a different method, both of which follow the rules of mathematical operations.\r<BR>\n" );
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document.write( "your solutions to this problem are:\r<BR>\n" );
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document.write( "when y = + sqrt(32), x = + 8/sqrt(32)<BR>\n" );
document.write( "when y = - sqrt(32), x = - 8/sqrt(32)<BR>\n" );
document.write( "when y = + sqrt(2), x = + 8/sqrt(2)<BR>\n" );
document.write( "when y = - sqrt(2), x = - 8/sqrt(2)\r<BR>\n" );
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document.write( "now we'll look at your second question.<BR>\n" );
document.write( "that question is:\r<BR>\n" );
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document.write( "2. $9000 were divided equally among a certain number of persons. Had there been 20 persons more, each would have got $160 less. Find the original number of persons.\r<BR>\n" );
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document.write( "let x = number of people<BR>\n" );
document.write( "let y = amount each person received originally\r<BR>\n" );
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document.write( "you get 2 equations.<BR>\n" );
document.write( "y = 9000/x ***** first equation<BR>\n" );
document.write( "(y-160) = 9000 / (x+20) ***** second equation\r<BR>\n" );
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document.write( "add 160 to both sides of the second equation to get:<BR>\n" );
document.write( "y = 9000 / (x+20) + 160\r<BR>\n" );
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document.write( "since 9000/x and 9000 / (x+20) + 160 are both equal to y, then they are both equal to each other, so you get:<BR>\n" );
document.write( "9000/x = 9000/(x+20) + 160<BR>\n" );
document.write( "multiply both sides of this equation by x * (x+20) to get:<BR>\n" );
document.write( "9000 * (x+20) = (9000 * x) + (160 * x * (x+20))<BR>\n" );
document.write( "simplify this equation to get:<BR>\n" );
document.write( "9000*x + 9000*20 = 9000*x + 160*(x^2+20x)<BR>\n" );
document.write( "simplify further to get:<BR>\n" );
document.write( "9000*x + 180000 = 9000*x + 160*x^2 + 3200*x<BR>\n" );
document.write( "subtract 9000*x and 180000 from both sides of the equation to get:<BR>\n" );
document.write( "0 = 9000*x - 9000*x + 160*x^2 + 3200*x - 180000<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "0 = 160*x^2 + 3200*x - 180000<BR>\n" );
document.write( "divide both sides of this equation by 160 to get:<BR>\n" );
document.write( "0 = x^2 + 20x - 1125<BR>\n" );
document.write( "since, if a = b, then b = a, this is the same as:<BR>\n" );
document.write( "x^2 + 20x - 1125 = 0<BR>\n" );
document.write( "factor this equation to get:<BR>\n" );
document.write( "(x + 45) * (x - 25) = 0<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = -45<BR>\n" );
document.write( "x = 25<BR>\n" );
document.write( "since the number of people can't be negative, then your only solution is:<BR>\n" );
document.write( "x = 25<BR>\n" );
document.write( "solve for y in the equation of y = 9000/x to get:<BR>\n" );
document.write( "y = 9000/25 = 360<BR>\n" );
document.write( "your solution appears to be:<BR>\n" );
document.write( "x = 25<BR>\n" );
document.write( "y = 360<BR>\n" );
document.write( "confirm by substituting these values in the original equations.\r<BR>\n" );
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document.write( "y = 9000/x becomes 360 = 9000/25 which becomes 360 = 360.<BR>\n" );
document.write( "solution is confirmed for first equation.\r<BR>\n" );
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document.write( "y-160 = 9000 / (x+20) becomes 360 - 160 = 9000 / 45 which becomes 200 = 200.<BR>\n" );
document.write( "solution is confirmed for second equation.\r<BR>\n" );
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document.write( "your solution is:\r<BR>\n" );
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document.write( "x = 25 which means that the original number of people was 25.<BR>\n" );
document.write( " </SPAN>\n" );
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