document.write( "Question 1106967: if rs =8y+2, st=2y+3, and rt=45 find the value of y \n" ); document.write( "

Algebra.Com's Answer #721992 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "if rs =8y+2, st=2y+3, and rt=45 find the value of y
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document.write( "We are given\r\n" );
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document.write( "rs = 8y+2,     (1)\r\n" );
document.write( "st = 2y+3.     (2)\r\n" );
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document.write( "Multiply equations (1) and (2) (both sides). You will get\r\n" );
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document.write( "    rt*s^2 = (8y+2)*(2y+3).   (3)\r\n" );
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document.write( "In the left side of (3),  replace rt by 45,  according to the condition.   You will get\r\n" );
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document.write( "    (8y+2)*(2y+3) = 45*s^2.   (4)\r\n" );
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document.write( "It is the quadratic equation.  Its right side is a positive number (for any triple (r,s,t)).\r\n" );
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document.write( "The equation has two different solutions for y: one solution in the interval (,)  and the other solution in the interval  (,). \r\n" );
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\n" ); document.write( "Now, the problem of this post has a HUGE UNCERTAINTY in its formulation.\r
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\n" ); document.write( "\n" ); document.write( "One can interpret the problem in these different ways:\r
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document.write( "1.  There are 4 unknowns and only 3 equations.  Hence, one can expect infinitely many solutions.  Then what the request \"find the value of y\" means ?\r\n" );
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document.write( "    a)  Does it mean to find some specific/special value ?\r\n" );
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document.write( "    b)  Does it mean to find at least one value of y ?\r\n" );
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document.write( "    c)  Does it mean to find the general solution or a general procedure for getting infinitely many solutions?\r\n" );
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document.write( "    d) what is really given in this problem ?\r\n" );
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document.write( "       Are r, s, t  given by their numerical values?\r\n" );
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document.write( "       Or we have only symbolic system of equations to be solved symbolically?\r\n" );
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\n" ); document.write( "\n" ); document.write( "By having so many options, I will restrict my contribution by these two considerations:\r
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document.write( "    a) find some specific/special values of r, t, s and y.\r\n" );
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document.write( "       Take y by an arbitrary way. For example, let y = 1.\r\n" );
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document.write( "       Then calculate  8y+2 = 10  and  2y+3 = 5.\r\n" );
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document.write( "       Next calculate s from (4): (8y+2)*(2y+3) = 10*5 = 50  ====>  45s^2 = 50  ====>  s^2 = 50/45 = 10/9 ====>  s =  = .\r\n" );
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document.write( "       Last step, determine r and t from\r\n" );
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document.write( "            rs = 8y+2 = 10  ====>  r =  =  = .\r\n" );
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document.write( "            st = 2y+3 = 5  ====>  t =  =  = .\r\n" );
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document.write( "       \r\n" );
document.write( "       Thus one special solution is  (r,t,s,y) = (, , , ).\r\n" );
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document.write( "    c)  Find the general solution/(general procedure) for getting infinitely many solutions.\r\n" );
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document.write( "        This general procedure is as follows:\r\n" );
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document.write( "            Take the value of s by an arbitrarily way;\r\n" );
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document.write( "            Find  \"y\"  from the quadratic equation (4);\r\n" );
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document.write( "            Having this value of  \"y\",  calculate  8y+2  and  2y+3;\r\n" );
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document.write( "            As the final step, calculate  r =   and  t = .\r\n" );
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document.write( "            Then the four numbers (r,s,t,y) are the solution to the system,\r\n" );
document.write( "            and this procedure provides infinitely many solutions = \"general solution\".\r\n" );
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