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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 (,) and the other solution in the interval (,).
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Now, the problem of this post has a HUGE UNCERTAINTY in its formulation.
One can interpret the problem in these different ways:
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 ?
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?
By having so many options, I will restrict my contribution by these two considerations:
a) find some specific/special values of r, t, s and y.
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 = = .
Last step, determine r and t from
rs = 8y+2 = 10 ====> r = = = .
st = 2y+3 = 5 ====> t = = = .
Thus one special solution is (r,t,s,y) = (, , , ).
c) Find the general solution/(general procedure) for getting infinitely many solutions.
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 = and t = .
Then the four numbers (r,s,t,y) are the solution to the system,
and this procedure provides infinitely many solutions = "general solution".