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Mathematica Algebra Equation Solving Wolfram Language

How to solve these simultaneous equations?

rr003035

rr003035 , University

Posted 10 years ago

Hello community, I am university student. I started to use Mathematica in few days ago. I want to solve the following simultaneous equations. I tried "Solve" and "NSolve", but I could not obtain answers. Do I make a mistake using these functions? Please tell me how to solve these equations. Which functions are proper to solve these equations? Sqrt[(-19.4 + x)^2 + (20.8 + y)^2 + (-11.4 + z)^2] + Sqrt[ x^2 + y^2 + z^2]=-10.496 Sqrt[(-16.5 + x)^2 + (29.2 + y)^2 + (-6.7 + z)^2] + Sqrt[ x^2 + y^2 + z^2]=-8.382 Sqrt[(-17.6 + x)^2 + (33.2 + y)^2 + (1.3 + z)^2] + Sqrt[ x^2 + y^2 + z^2] =-8.938

POSTED BY: rr003035

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Frank Kampas

Frank Kampas, Physicist at Large Consulting

Posted 10 years ago

For setting up equations, use == , not =. However, they don't appear to have a solution. NSolve[{Sqrt[(-19.4 + x)^2 + (20.8 + y)^2 + (-11.4 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -10.496, Sqrt[(-16.5 + x)^2 + (29.2 + y)^2 + (-6.7 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -8.382, Sqrt[(-17.6 + x)^2 + (33.2 + y)^2 + (1.3 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -8.938}, {x, y, z}] {}

For setting up equations, use == , not =. However, they don't appear to have a solution.

NSolve[{Sqrt[(-19.4 + x)^2 + (20.8 + y)^2 + (-11.4 + z)^2] + 
    Sqrt[x^2 + y^2 + z^2] == -10.496, 
  Sqrt[(-16.5 + x)^2 + (29.2 + y)^2 + (-6.7 + z)^2] + 
    Sqrt[x^2 + y^2 + z^2] == -8.382, 
  Sqrt[(-17.6 + x)^2 + (33.2 + y)^2 + (1.3 + z)^2] + 
    Sqrt[x^2 + y^2 + z^2] == -8.938}, {x, y, z}]

{}

POSTED BY: Frank Kampas

Nasser M. Abbasi

Nasser M. Abbasi, student

Posted 10 years ago

You can start by trying FindRoot eq1 = Sqrt[(-19.4 + x)^2 + (20.8 + y)^2 + (-11.4 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -10.496; eq2 = Sqrt[(-16.5 + x)^2 + (29.2 + y)^2 + (-6.7 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -8.382; eq3 = Sqrt[(-17.6 + x)^2 + (33.2 + y)^2 + (1.3 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -8.938; FindRoot[{eq1, eq2, eq3}, {x, 1}, {y, 1}, {z, 1}] {x -> 5.46311, y -> -2.52191, z -> -0.822461} If you can setup your system as `Ax=b`, you can also try `LinearSolve` which obtain least squares approximation of solution. But your equations are not linear, so this would not work in this case. There might be other methods, but if NSolve fails, that is what I try next (btw, use the `Reals` option with `NSolve` and `Solve` helps sometimes finding a solution, but not in this case). May be someone else will have better answer for you.

POSTED BY: Nasser M. Abbasi

Frank Kampas

Frank Kampas, Physicist at Large Consulting

Posted 10 years ago

FindRoot issues a line search tolerance warning and doesn't find a very good solution: eq1 = Sqrt[(-19.4 + x)^2 + (20.8 + y)^2 + (-11.4 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -10.496; eq2 = Sqrt[(-16.5 + x)^2 + (29.2 + y)^2 + (-6.7 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -8.382; eq3 = Sqrt[(-17.6 + x)^2 + (33.2 + y)^2 + (1.3 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -8.938; sln = FindRoot[{eq1, eq2, eq3}, {x, 1}, {y, 1}, {z, 1}] ({eq1, eq2, eq3} /. a_ == b_ -> a - b ) /. sln {42.602, 44.29, 48.0062}

FindRoot issues a line search tolerance warning and doesn't find a very good solution:

eq1 = Sqrt[(-19.4 + x)^2 + (20.8 + y)^2 + (-11.4 + z)^2] + 
    Sqrt[x^2 + y^2 + z^2] == -10.496;
eq2 = Sqrt[(-16.5 + x)^2 + (29.2 + y)^2 + (-6.7 + z)^2] + 
    Sqrt[x^2 + y^2 + z^2] == -8.382;
eq3 = Sqrt[(-17.6 + x)^2 + (33.2 + y)^2 + (1.3 + z)^2] + 
    Sqrt[x^2 + y^2 + z^2] == -8.938;
sln = FindRoot[{eq1, eq2, eq3}, {x, 1}, {y, 1}, {z, 1}]

({eq1, eq2, eq3} /. a_ == b_ -> a - b ) /. sln
{42.602, 44.29, 48.0062}

POSTED BY: Frank Kampas

David Keith

Posted 10 years ago

Reduce thinks there's no solution: eqs = {Sqrt[(-19.4 + x)^2 + (20.8 + y)^2 + (-11.4 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -10.496, Sqrt[(-16.5 + x)^2 + (29.2 + y)^2 + (-6.7 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -8.382, Sqrt[(-17.6 + x)^2 + (33.2 + y)^2 + (1.3 + z)^2] + Sqrt[x^2 + y^2 + z^2] == -8.938} In[3]:= Reduce[eqs] During evaluation of In[3]:= Reduce::ratnz: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. >> Out[3]= False

Reduce thinks there's no solution:

eqs = {Sqrt[(-19.4 + x)^2 + (20.8 + y)^2 + (-11.4 + z)^2] + 
    Sqrt[x^2 + y^2 + z^2] == -10.496, 
  Sqrt[(-16.5 + x)^2 + (29.2 + y)^2 + (-6.7 + z)^2] + 
    Sqrt[x^2 + y^2 + z^2] == -8.382, 
  Sqrt[(-17.6 + x)^2 + (33.2 + y)^2 + (1.3 + z)^2] + 
    Sqrt[x^2 + y^2 + z^2] == -8.938}  
  In[3]:= Reduce[eqs]

During evaluation of In[3]:= Reduce::ratnz: Reduce was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. >>

Out[3]= False

POSTED BY: David Keith