# Can't solve system of six equations in five variables

Here's an equations from mechanics:

eq1 := Xa2 + Xb2 - (m1 + m2)*g*Sin[φ] + Fc == 0
eq2 := Ya2 + Yb2 - (m1 + m2)*g*Cos[φ] - Fe - F1 + Fr*Sin[α] == 0
eq3 := Za2 + Fr*Cos[α] == 0
eq4 := Yb2*2*R - F1*R - Fe*(R - x*Cos[α]) + Fr*R*Sin[α] - (R*m1 + m2*(Rx*Cos[α]))*g*Cos[φ] == 0
eq5 := -Xb2*2*R - Fc*(R - x*Cos[α]) - (R*m1 + m2*(R - x*Cos[α]))*g*Sin[φ] == 0
eq6 := Mvr - Fc*x*Sin[α] - g*Sin[φ]*(4*R/(3*π)*m1 + m2*x*Sin[α]) == 0
Solve[{eq1, eq2, eq3, eq4, eq5, eq6}, {Xa2, Xb2, Ya2, Yb2, Za2}]


Clear["Global*"]

eq1 := Xa2 + Xb2 - (m1 + m2)*g*Sin[φ] + Fc == 0;
eq2 := Ya2 + Yb2 - (m1 + m2)*g*Cos[φ] - Fe - F1 +
Fr*Sin[α] == 0;
eq3 := Za2 + Fr*Cos[α] == 0;
eq4 := Yb2*2*R - F1*R - Fe*(R*x*Cos[α]) +
Fr*R*Sin[α] - (R*m1 + m2*(Rx*Cos[α]))*g*Cos[φ] ==
0;
eq5 := -Xb2*2*R - Fc*(R - x*Cos[α]) -
(R*m1 + m2*(R - x*Cos[α]))*g*Sin[φ] == 0;
eq6 := Mvr - Fc*x*Sin[α] -
g*Sin[φ]*(4*R/(3*π)*m1 + m2*x*Sin[α]) == 0;

Solve[{eq1, eq2, eq3, eq4, eq5, eq6}, {Xa2, Xb2, Ya2, Yb2, Za2}]

(* {} *)


The system is overdetermined. Looking at the Options for Solve

Options[Solve]

(* {Cubics -> Automatic, GeneratedParameters -> C, InverseFunctions -> Automatic,
MaxExtraConditions -> 0, Method -> Automatic, Modulus -> 0,
Quartics -> Automatic, VerifySolutions -> Automatic,
WorkingPrecision -> ∞} *)


Then specifically at the option MaxExtraConditions

?MaxExtraConditions


(sol = Solve[{eq1, eq2, eq3, eq4, eq5, eq6}, {Xa2, Xb2, Ya2, Yb2, Za2},
MaxExtraConditions -> 1][[1]]) // Column


Each ConditionalExpression has the same condition

assume = sol[[All, -1, -1]] // Union

(* {-3 Mvr π + 3 Fc π x Sin[α] + 4 g m1 R Sin[φ] +
3 g m2 π x Sin[α] Sin[φ] == 0} *)


To eliminate the condition, you can either Simplify using this assumption or use Normal

(soln = sol // Normal) // Column


To solve for Xa2 and Ya2 in terms of Xb2 and Yb2

(soln2 = Solve[soln[[1 ;; 4]] /. Rule :> Equal, {Xa2, Ya2}, {R, x}][[1]] //
Simplify) // Column


• After changing the third equation to eq3 := Za2 - Fr*Cos[α] == 0` and executing all of the code, I do not get an error message. What error message are you getting? Commented Aug 25, 2019 at 16:12