solving systems of equations returns [] Redux

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I can not reproduce your problem on 7.4.beta2:

sage: var('x' ,'y' ,'z', 'lam')
(x, y, z, lam)
sage: F = x*y*z - lam * (x*y*z-9) 
sage: F
x*y*z - (x*y*z - 9)*lam
sage: dFx=diff(F,x)
sage: dFy=diff(F,y)
sage: dFz=diff(F,z)
sage: dFlam=diff(F,lam)
sage: dFz
-lam*x*y + x*y
sage: solve([dFx==0,dFy==0,dFz==0,dFlam==0],x,y,z,lam)
[[x == 9/(r1*r2), y == r1, z == r2, lam == 1]]

This seems to work in Sage 7.3 (the latest public release).

Not sure what change between Sage 7.2 and Sage 7.3 made this work.

One could try to check which ticket is responsible among those merged in Sage 7.3.

http://www.sagemath.org/changelogs/sa...

$ sage -v
SageMath version 7.3, Release Date: 2016-08-04
$ sage -q
sage: x, y, z, lam = var('x' ,'y' ,'z', 'lam')
sage: F = x*y*z - lam * (x*y*z-9); F
x*y*z - (x*y*z - 9)*lam
sage: dFx = diff(F, x)
sage: dFy = diff(F, y)
sage: dFz = diff(F, z)
sage: dFlam = diff(F, lam)
sage: dFx, dFy, dFz, dFlam
(-lam*y*z + y*z, -lam*x*z + x*z, -lam*x*y + x*y, -x*y*z + 9)
sage: solve([dFx == 0, dFy == 0, dFz == 0, dFlam == 0], x, y, z, lam)
[[x == 9/(r1*r2), y == r1, z == r2, lam == 1]]
sage: