# How to solve a system of equations?

Using the below sets of equations, I want to find the expression for ζ, η and eg using Mathematica.

\begin{aligned}\sqrt{\eta} \zeta = g \frac{1-\lambda_{\text{ch}}^2}{1-g^2 \lambda_{\text{ch}}^2} \sqrt{T} \lambda, \hspace{24 mm}\\ \frac{\eta \epsilon^g}{2+\eta \epsilon^g} =g^2 \frac{T \epsilon}{2+T \epsilon},\hspace{36 mm}\\ \frac{\eta\left[\zeta^2\left(2-\epsilon^g\right)+\epsilon^g\right]}{2-\zeta^2\left[2+\eta\left(\epsilon^g-2\right)\right]+\eta \epsilon^g}=g^2 \frac{T\left[\lambda^2(2-\epsilon)+\epsilon\right]}{2-\lambda^2[2+T(\epsilon-2)]+T \epsilon}\end{aligned}

\begin{aligned}\lambda_{\text{ch}}^2&=\frac{T\epsilon}{2+T\epsilon}\end{aligned}

The above system of equation should be reduced to: \begin{aligned}\zeta & =\lambda \sqrt{\frac{\left(g^{2}-1\right)(\epsilon-2) T-2}{\left(g^{2}-1\right) \epsilon T-2}},\\ \eta &=\frac{g^{2} T}{\left(g^{2}-1\right) T\left[\frac{1}{4}\left(g^{2}-1\right)(\epsilon-2) \epsilon T-\epsilon+1\right]+1}, \\ \epsilon^{g}&=\epsilon+\frac{1}{2}\left(g^{2}-1\right)(2-\epsilon) \epsilon T, \hspace{43 mm} \end{aligned} Could you please guide me on how to achieve this?

Code:

Solve[Sqrt[η] ζ == g λ Sqrt[T] (1 - λ^2)/(1 - g^2 λ^2) &&
η eg/(2 + η eg) ==  g^2 T ϵ/(2 + T ϵ) &&
η (ζ^2 (2 - eg) + eg)/(2 - ζ^2 (2 + η (eg - 2)) + η eg) ==
g^2 T (λ^2 (2 - ϵ) + ϵ)/(2 - λ^2 (2 + T (ϵ - 2)) + T ϵ), {ζ, η, eg}]

• What happens when you run your Solve expression? Commented Aug 1, 2023 at 12:26
• I'm not getting any response, and the process seems to be stuck in a running status Commented Aug 1, 2023 at 13:04
• Check your code, it has typos. Commented Aug 1, 2023 at 13:41
• Does the context which gave rise to these equations indicate any constraints on the variables and/or parameters? Are any Positive? NonNegative? Bounded? Using the constraints as assumptions might help. Commented Aug 2, 2023 at 5:49
• Constraints on the variables: 0 ≤ T ≤ 1, ∈ = 2 * NB / T < 2, g > 1 and 𝜆^2 < 1 Commented Aug 3, 2023 at 4:49

Simply include the constraint condition cond and limit it to Reals. After that, everything becomes straightforward.

$$\lambda_{ch}$$ is replaced because it is a intermediate variable, which doesn't appear in the final solution.

P.S. Convert my code to StandardForm would be more readable.

Clear["Global*"]

eqs = {Sqrt[\[Eta]] \[Zeta] ==
g \[Lambda] Sqrt[T] (1 - Subscript[\[Lambda], ch]^2)/(
1 - g^2 Subscript[\[Lambda], ch]^2) ,
(\[Eta] eg)/(2 + \[Eta] eg) ==
g^2 (T \[Epsilon])/(2 + T \[Epsilon]) ,
( \[Eta] (\[Zeta]^2 (2 - eg) + eg))/(
2 - \[Zeta]^2 (2 + \[Eta] (eg - 2)) + \[Eta] eg) ==
g^2 (T (\[Lambda]^2 (2 - \[Epsilon]) + \[Epsilon]))/(
2 - \[Lambda]^2 (2 + T (\[Epsilon] - 2)) + T \[Epsilon])};

(* replace lambda_ch *)
eqsf = FullSimplify[
eqs /. {Subscript[\[Lambda], ch]^2 -> (T \[Epsilon])/(
2 + T \[Epsilon])}]

cond = {0 <= T <= 1, \[Epsilon] < 2, g > 1, -1 <= \[Lambda] <= 1};

sol=Solve[Join[eqsf , cond], {\[Zeta], \[Eta], eg}, Reals]

FullSimplify[sol]

eqsf /. sol // FullSimplify

• Could you add some commentary to explain your approach? Commented Aug 4, 2023 at 8:59
• @Peace Wang: Yes, this is the final expression I was aiming for. Thank you very much for your help Commented Aug 5, 2023 at 7:02
• If my answer is helpful, you can do me a favor to accept the answer Commented Aug 5, 2023 at 9:08
\$Version

(* "13.3.0 for Mac OS X ARM (64-bit) (June 3, 2023)" *)

Clear["Global*"]

eq1 = Sqrt[η] ζ == g λ Sqrt[T] (1 - λ^2)/(1 - g^2 λ^2);
eq2 = η eg/(2 + η eg) == g^2 T ϵ/(2 + T ϵ);
eq3 = η (ζ^2 (2 - eg) +
eg)/(2 - ζ^2 (2 + η (eg - 2)) + η eg) ==
g^2 T (λ^2 (2 - ϵ) + ϵ)/(2 - λ^2 (2 + T (ϵ - 2)) + T ϵ);


Solve the equations in steps

solζ = Solve[eq1, ζ][[1]];

solη = Solve[eq2, η][[1]];

eq4 = eq3 /. solζ /. solη // Simplify;

soleg = Solve[eq4, eg][[1]] // FullSimplify;

solη = solη /. soleg // FullSimplify;

solζ = solζ /. solη;

sol = {soleg, solη, solζ} // Flatten

(* {eg -> -((ϵ ((-1 +
g^2)^2 T^2 ϵ (-1 + λ^2)^2 (-ϵ + (-2 + ϵ) λ^2) -
4 (-1 + g^2) T (-1 + λ^2)^2 (-ϵ + (-1 + ϵ) λ^2) +
4 λ^2 (3 -
2 g^2 + (-3 + g^4) λ^2 + λ^4)))/(2 (-2 + (-1 +
g^2) T ϵ) λ^2 (-1 + λ^2)^2)),
η -> (4 g^2 T λ^2 (-1 + λ^2)^2)/((-1 +
g^2)^2 T^2 ϵ (-1 + λ^2)^2 (-ϵ + (-2 + ϵ) λ^2) -
4 (-1 + g^2) T (-1 + λ^2)^2 (-ϵ + (-1 + ϵ) λ^2) +
4 λ^2 (3 - 2 g^2 + (-3 + g^4) λ^2 + λ^4)),
ζ -> (g Sqrt[T] λ (-1 + λ^2))/(2 (-1 +
g^2 λ^2) √((g^2 T λ^2 (-1 + λ^2)^2)/((-1 \
+ g^2)^2 T^2 ϵ (-1 + λ^2)^2 (-ϵ + (-2 + ϵ) λ^2) -
4 (-1 + g^2) T (-1 + λ^2)^2 (-ϵ + (-1 + ϵ) λ^2) +
4 λ^2 (3 - 2 g^2 + (-3 + g^4) λ^2 + λ^4))))} *)


Verifying the solution,

{eq1, eq2, eq3} /. sol // Simplify

(* {True, True, True} *)

• This is not the simplified expressions that I am trying to obtain. Nevertheless, Thank you very much for your valuable contribution and efforts. Commented Aug 1, 2023 at 14:29