# solving systems of equations returns [] Redux

I searched the wiki and found the "solve always returns []" question. But it doesn't help me. I will admit that I am a complete neophyte with Sagemath and have been playing with version 7.2 on my windows machine.

I was trying to do something very simple: solve a basic Lagrange Multiplier problem, so I defined the following:

• var('x' ,'y' ,'z', 'lam')
• var('F')
• F = xyz - lam * (xyz-9)
• var('dFx', 'dFy', 'dFz', 'dFlam')
• dFx=diff(F,x)
• dFy=diff(F,y)
• dFz=diff(F,z)
• dFlam=diff(F,lam)
• solve([dFx==0,dFy==0,dFz==0,dFlam==0],x,y,z,lam)

but the solve returns []. It should return something like x==3, y==3, z==3, shouldn't it? Is there an alternative way to approach this?

Thanks.

BobM

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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]]

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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:
`
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