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

Answer

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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

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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

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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