# Trying to solve a system of equation

I am trying to solve 3 equations 3 unknown system on Mathematica in order to get numerical results for my three variables x,y,z but the software is always on the Running... status and never gives a result. (It does not give an error message neither. Mathematica stays actionless and after I smust reinitialize it.)

 parametres = {
σ -> 1.5
, α -> 0.3
, β -> 0.6
, ρ -> 0.02
, δ -> 0.05
, ϕ -> 0.8
};


After the calibration, I use

sol11 = Solve[
{
((α y - ρ) (α - 1))/(σ (α - β - 1)) == x
, (β x)/(α - 1) + y^(α - 1) == z
, x ((β (1 - σ))/(1 - α) + (βy^α)/(ϕ z)) == ρ
}
, \{x, y, z}] /. parametres // Simplify


What could be the problem ? Does it mean that there does not exist a result ?

• Your code is currently trying to find a symbolic result first, and then substituting the numerical values into it. You might have better luck if you do the parameter substitution first: Solve[{...}/.parametres, {x, y, z}]. Alternatively, if that doesn't work either, you can try FindRoot. – MarcoB Aug 21 '15 at 12:24
• @MarcoB Thanks so much. Solve did not work but FindRoot worked perfectly ! – optimal control Aug 21 '15 at 17:38

parametres = {σ -> 1.5, α -> 0.3, β -> 0.6, ρ ->
0.02, δ -> 0.05, ϕ -> 0.8} // Rationalize;


Your third equation contains (βy^α) there needs to be a space as in (β y^α)

eqns = {((α y - ρ) (α -
1))/(σ (α - β - 1)) ==
x, (β x)/(α - 1) + y^(α - 1) == z,
x ((β (1 - σ))/(1 - α) + (β \
y^α)/(ϕ z)) == ρ} /. parametres // Simplify;


Restrict the domain to Reals

sol11 = Solve[eqns, {x, y, z}, Reals] // N


{{x -> 0.082946, y -> 0.836879, z -> 1.06166}}

Since Rationalize was used to provide exact numbers for the parameters, Solve provides an exact solution in terms of Root objects. //N was used to convert these to numerical values. Verifying the result:

eqns /. sol11[]

{True, True, True}

• Thanks for the clear presentation. I did not think in this way. – optimal control Aug 21 '15 at 17:39

I guess the FindInstance function does the job:

FindInstance[{((α y - ρ) (α - 1))/(σ (α - β - 1)) == x
, (β x)/(α - 1) + y^(α - 1) == z
, x ((β (1 - σ))/(1 - α) + (β y^α)/(ϕ z)) == ρ}
/. parametres, {x, y, z}, Reals]


{{x -> 0.082946, y -> 0.836879, z -> 1.06166}}

• Thanks for the use of this command to which I was not familiar. – optimal control Aug 21 '15 at 17:39