AIMO Question which is bugging me (1 Viewer)

[PC]

Find x + y where x and y are non-zero solutions of the system of equations:
y²x = 15x² + 17xy + 15y²
x²y = 20x² + 3y²

I'm fairly sure the solution is x=19 and y=95. Just don't know how to get there.

 

[darkliight]

Just algebra I think...

Multiply the first equation by x and the second by y. This gives,
y²x² = 15x³ + 17x²y + 15y²x
x²y² = 20x²y + 3y³.

That is,
15x³ + 17x²y + 15y²x = 20x²y + 3y³.

Rearranging gives
-x²(y-5x) = y²(y-5x).

So either x = y = 0 (which we don't want) or y - 5x = 0 => y = 5x.
Sub this into the second equation to get
5x³ = 95x² => x²(5x-95) = 0 => x = 19 => y = 95.

The required sum follows.

 

[tehcreativ1]

Just algebra I think...

Multiply the first equation by x and the second by y. This gives,
y²x² = 15x³ + 17x²y + 15y²x
x²y² = 20x²y + 3y³.

That is,
15x³ + 17x²y + 15y²x = 20x²y + 3y³.

Rearranging gives
-x²(y-5x) = y²(y-5x).

So either x = y = 0 (which we don't want) or y - 5x = 0 => y = 5x.
Sub this into the second equation to get
5x³ = 95x² => x²(5x-95) = 0 => x = 19 => y = 95.

The required sum follows.

Yup, darkliight's solution is awesome =D just a note that the first step (multiplying through to get equivalent expressions) can only be done coz x and y are nonzero. otherwise, yay for darkliight and yay for aimo ^^

 

[darkliight]

Hmm, since x = 0 if and only if y = 0, does it matter if x, y are nonzero in the first step?