document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=969357>Question 969357</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A tourist sailed against the current on a river for 6 km, and then he sailed in a lake for 15 km. In the lake he sailed for 1 hour longer than he sailed in the river. Knowing that the current of the river is 2 km/hour, find the speed of the boat while it is traveling in the lake<BR>\n" );
document.write( "6/(x-2) = 15/x +1<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 #592285 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=592285\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> it looks like you had the right idea to start with but maybe didn't know how to finish it.\r<BR>\n" );
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document.write( "the general formula is r * t = d\r<BR>\n" );
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document.write( "r is the rate of speed.<BR>\n" );
document.write( "t is the time.<BR>\n" );
document.write( "d is the distance.\r<BR>\n" );
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document.write( "the formula in the river becomes:\r<BR>\n" );
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document.write( "(r-2) * t = 6\r<BR>\n" );
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document.write( "the formula in the lake becomes:\r<BR>\n" );
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document.write( "r * (t + 1) = 15\r<BR>\n" );
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document.write( "you have 2 equations tht need to be solved simultaneously because the same value of r and the same value of t applies to both equations.\r<BR>\n" );
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document.write( "those equation are:\r<BR>\n" );
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document.write( "(r-2) * t = 6<BR>\n" );
document.write( "r * (t + 1) = 15\r<BR>\n" );
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document.write( "there are various ways to solve two equations simultaneously.<BR>\n" );
document.write( "we'll do it the way that is shown below:<BR>\n" );
document.write( "just keep in mind that this isn't the only way that simultensous equations can be solved.\r<BR>\n" );
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document.write( "there substitution and elimination and graphing.<BR>\n" );
document.write( "those are the big three.<BR>\n" );
document.write( "this way is by elimination.<BR>\n" );
document.write( "you can also solve both equations for the same variable and then set those equations equal to each other.<BR>\n" );
document.write( "the whole idea is to eliminate one of the variables so the other variable can be solved for.<BR>\n" );
document.write( "once that variable is solved for, you can use that value to solve for the other variable.\r<BR>\n" );
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document.write( "simplify these equations to get:\r<BR>\n" );
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document.write( "rt - 2t = 6<BR>\n" );
document.write( "rt + r = 15\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( "r + 2t = 9\r<BR>\n" );
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document.write( "solve for r to get r = 9 - 2t.\r<BR>\n" );
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document.write( "go back to the first original equation and replace r with 9 - 2t to get (r-2)*t = 6 to become (9 - 2t - 2) * t = 6\r<BR>\n" );
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document.write( "simplify that to get 9t - 2t^2 - 2t = 6\r<BR>\n" );
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document.write( "simplify further to get 7t - 2t^2 = 6\r<BR>\n" );
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document.write( "add 2t^2 to both sides of the equation and subtract 7t from both sides of the equation to get 0 = 2t^2 - 7t + 6 which is the same as 2t^2 - 7t + 6 = 0\r<BR>\n" );
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document.write( "factor that quadratic equation to get t - 2 or t = 3/2.\r<BR>\n" );
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document.write( "solve for r in the first original equation and you will get r = 5 or r = 6.\r<BR>\n" );
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document.write( "solve for r in the second original equation and you will get r = 5 or r = 6 as well.\r<BR>\n" );
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document.write( "your solutions are:\r<BR>\n" );
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document.write( "when t = 2, r = 5<BR>\n" );
document.write( "when t = 3/2, r = 6\r<BR>\n" );
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document.write( "both solutions look good.\r<BR>\n" );
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document.write( "use them in both original equations and those equations should be true.\r<BR>\n" );
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document.write( "when t = 2 and r = 5, both original equations become:\r<BR>\n" );
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document.write( "(r-2)*t = 6 becomes (5-2)*2 = 6 which becomes 3*2 = 6 which becomes 6 = 6 which is true.\r<BR>\n" );
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document.write( "r*(t+1) = 15 becomes 5*(2+1) = 15 which becomes 15 = 15 which is true.\r<BR>\n" );
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document.write( "when t = 1.5 and r = 6, both original equations becomes:\r<BR>\n" );
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document.write( "(r-2)*t = 6 becomes (6-2)*1.5 = 6 which becomes 4*1.5 = 6 which becomes 6 = 6 which is true.\r<BR>\n" );
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document.write( "r*(t+1) = 15 becomes 6*(1.5+1) = 15 which becomes 6*2.5 = 15 which becomes 15 = 15 which is true.\r<BR>\n" );
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document.write( "both solutions are good.\r<BR>\n" );
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document.write( "the problem statement was:\r<BR>\n" );
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document.write( "find the speed of the boat while it is traveling in the lake.\r<BR>\n" );
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document.write( "the answer is that the speed of the boat when it is traveling in the lake is either 5 or 6 kmph.\r<BR>\n" );
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document.write( "it could be either one, unless i did something drastically wrong.\r<BR>\n" );
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document.write( "i also solved it graphically and got the same result.\r<BR>\n" );
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document.write( "there are two possible solutions as i described above.\r<BR>\n" );
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document.write( "to graph these equations, let y = t and let x = r and solve for y in both eqautions.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "y = 6/(x-2) and y = 15/x - 1\r<BR>\n" );
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document.write( "these are pretty close to what you had, except you had 15/x + 1 which was incorrect.\r<BR>\n" );
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document.write( "the graph would look something like this:\r<BR>\n" );
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document.write( "it's hard to see on this graph, but there are 2 intersection points at x = 6 and x = 6.\r<BR>\n" );
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document.write( "when x = 5, the value of y is 2.\r<BR>\n" );
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document.write( "when x = 6, the value of y is 1.5\r<BR>\n" );
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document.write( "the horizontal lines are the values of y = 2 and y = 1.5.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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