This is my first question here, so hopefully I will be able to explain my problem in a coherent way. My ultimate question is: do you see any way I can simplify the following system so I can have a more intuitive solution to it? Let me explain in details what I mean:I want to characterize x,y,z that solve the following system of non-linear equations:Δx=θΔzΔyΔy−θΔz (EQ1)Δx+Δy+Δz=Y― (EQ2)θ(Δz)2+(Δy)2=U― (EQ3)whereΔx=x−x―Δy=y−y―Δz=z−z―and 0&lt;θ&lt;1,Y―,U―,x―,y― and z― are parameters.The way I've approached this was to parameterize (EQ2):Δy+Δz=αsuch that α=Y―−ΔxFor θ=0.8,x―=y―=z―=0.5 and Y―=−0.5 and U―=0.2 this parametric system would look like the following (I apologize I don't have reputation to embed the photo here):The parametric systemI can characterize Δy and Δz as a function of α:Δy=θα±U―(1+θ)−θα21+θΔz=α∓U―(1+θ)−θα21+θMy problem arrives in the characterization of α using (EQ1). Although we can show such an α exists, it is far from an intuitive closed-form characterization, as α solves:±(Y―−α)(1−θ)U―(1+θ)−θα2θα±U―(1+θ)−θα2=θ(α∓U―(1+θ)−θα2)Am I using the correct approach to this problem? Or do you see anyway I could simplify this? Thank you!

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This is my first question here, so hopefully I will be able to explain my problem in a coherent way. My ultimate question is: do you see any way I can simplify the following system so I can have a more intuitive solution to it? Let me explain in details what I mean:I want to characterize x,y,z that solve the following system of non-linear equations:Δx=θΔzΔyΔy−θΔz     (EQ1)Δx+Δy+Δz=Y―     (EQ2)θ(Δz)2+(Δy)2=U―     (EQ3)whereΔx=x−x―Δy=y−y―Δz=z−z―and 0&amp;lt;θ&amp;lt;1,Y―,U―,x―,y― and z― are parameters.The way I&#039;ve approached this was to parameterize (EQ2):Δy+Δz=αsuch that α=Y―−ΔxFor θ=0.8,x―=y―=z―=0.5 and Y―=−0.5 and U―=0.2 this parametric system would look like the following (I apologize I don&#039;t have reputation to embed the photo here):The parametric systemI can characterize Δy and Δz as a function of α:Δy=θα±U―(1+θ)−θα21+θΔz=α∓U―(1+θ)−θα21+θMy problem arrives in the characterization of α using (EQ1). Although we can show such an α exists, it is far from an intuitive closed-form characterization, as α solves:±(Y―−α)(1−θ)U―(1+θ)−θα2θα±U―(1+θ)−θα2=θ(α∓U―(1+θ)−θα2)Am I using the correct approach to this problem? Or do you see anyway I could simplify this? Thank you!

Kathryn Duggan · Accepted Answer

I will reason in terms of the Δ rather than the non-centered x,y,z.  The first equation, which can be written  ΔxΔy=θΔz(Δx+Δy)  is that of a cone with apex at the origin. The second is a plane and the third and elliptic cylindre of vertical axis. The latter two define an ellipse, which you can represent by the parametric equations  Δy=U―1−u21+u2  Δz=U―θ2u1+u2  Δx=Y―−Δy−Δz.  (The rational expressions can be obtained from the trigonometric representation.)  Plugging these expressions in the first equation, you will get a quartic equation in u, which can have zero, two or four solutions (intersection points of the ellipse and the cone).  It is unlikely that there will be a simple expression for these solutions, even though there are only three independent parameters and the quartic equation is not fully general.

Elena Ray · Accepted Answer

a=θbcb−θc  a+b+c=−12  θc2+b2=15  a=x−12  b=y−12  c=z−12

This is my first question here, so hopefully I will be able to explain my problem in a coherent way.

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New Questions in Algebra I