Question 1106967
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if rs =8y+2, st=2y+3, and rt=45 find the value of y
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We are given

rs = 8y+2,     (1)
st = 2y+3.     (2)


Multiply equations (1) and (2) (both sides). You will get

    rt*s^2 = (8y+2)*(2y+3).   (3)


In the left side of (3),  replace rt by 45,  according to the condition.   You will get

    (8y+2)*(2y+3) = 45*s^2.   (4)


It is the quadratic equation.  Its right side is a positive number (for any triple (r,s,t)).

The equation has two different solutions for y: one solution in the interval ({{{-infinity}}},{{{-3/2}}})  and the other solution in the interval  ({{{-1/4}}},{{{infinity}}}). 
</pre>

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Now, the problem of this post has a <U>HUGE UNCERTAINTY</U> in its formulation.


One can interpret the problem in these different ways:


<pre>
1.  There are 4 unknowns and only 3 equations.  Hence, one can expect infinitely many solutions.  Then what the request "find <U>the</U> value of y" means ?

    a)  Does it mean to find some specific/special value ?

    b)  Does it mean to find at least one value of y ?

    c)  Does it mean to find the general solution or a general procedure for getting infinitely many solutions?

    d) what is really given in this problem ?

       Are r, s, t  given by their numerical values?

       Or we have only symbolic system of equations to be solved symbolically?
</pre>

By having so many options, I will restrict my contribution by these two considerations:


<pre>
    a) <U>find some specific/special values of r, t, s and y.</U>


       Take y by an arbitrary way. For example, let y = 1.

       Then calculate  8y+2 = 10  and  2y+3 = 5.

       Next calculate s from (4): (8y+2)*(2y+3) = 10*5 = 50  ====>  45s^2 = 50  ====>  s^2 = 50/45 = 10/9 ====>  s = {{{sqrt(10/9)}}} = {{{sqrt(10)/3}}}.

       Last step, determine r and t from

            rs = 8y+2 = 10  ====>  r = {{{10/s}}} = {{{10/((sqrt(10)/3))}}} = {{{3*sqrt(10)}}}.

            st = 2y+3 = 5  ====>  t = {{{5/s}}} = {{{5/((sqrt(10)/3))}}} = {{{(3*sqrt(10))/2}}}.

       
       Thus one special solution is  (r,t,s,y) = ({{{3*sqrt(10)}}}, {{{(3*sqrt(10))/2}}}, {{{sqrt(10)/3}}}, {{{1}}}).


    c)  <U>Find the general solution/(general procedure) for getting infinitely many solutions</U>.

        This general procedure is as follows:

            Take the value of s by an arbitrarily way;

            Find  "y"  from the quadratic equation (4);

            Having this value of  "y",  calculate  8y+2  and  2y+3;

            As the final step, calculate  r = {{{(8y+2)/s}}}  and  t = {{{(2y+3)/s}}}.

            Then the four numbers (r,s,t,y) are the solution to the system,
            and this procedure provides infinitely many solutions = "general solution".
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