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Solve a generic set of equations (Posted on 2017-09-03) Difficulty: 3 of 5
What couples of numbers satisfy the following set of equations:

x^2+xy=t
y^2+xy=t*k
?

List all the qualifying couples.

Please verify for t=20 & k=2

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution (spoiler) Comment 1 of 1
Answer
------
For any given t and k

a) if t=0 then there are an infinite number of solutions of the form x = -y
b) if t <> 0, then the solutions (x,y) are 
    ( sqrt(t/(1+k)), k*sqrt(t/(1+k)) ) and ( -sqrt(t/(1+k)), -k*sqrt(t/(1+k)) )
    
Method
------
y(x+y) = tk
x(x+y) = t

If t <> 0 then divide the 1st equation by the 2nd, giving

y/x = k

substitute y = xk into the 2nd equation, giving

x(x+xk) = t
x^2* (1+k) = t
x = +/- sqrt(t/(1+k))

Requested verification
----------------------
If t = 20 and k = 2
then x = +/- sqrt(20/3)
If x is positive, then y = 2(sqrt(20/3))

x^2 + xy = 20/3 + 40/3 = 20 = t
y^2 + xy = 80/3 + 40/3 = 40 = tk

same if x and y are negative

  Posted by Steve Herman on 2017-09-03 11:56:51
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