How to solve a system of equations?

My assignment is to take the vapor compositions (yD=[0.3291 0.529 0.1418]) use these equations

x2*gamma2*P2sat=y2*1
x3*gamma3*P3sat=y3*1
x1*gamma1*P1sat+x2*gamma2*P2sat+x3*gamma3*P3sat=1
x1+x2+x3=1

and solve for x1, x2, x3 and T. I cannot figure out how to use fsolve to find these values.

Please help I have worked on this for 2 days and I do not understand what I am doing wrong.

Below are the function handles to determine the gamma and Psat values.

Thank you

    %Ternary Distillation Day 2

clear
clc

R=1.987;    %cal/K mol
P=1;        %atm

%Used the composition from trial 2 since it was closest to equilibrium

%Distillate vapor composition
yD=[0.3291 0.529 0.1418];

%molar volume equation equation found from ploting the points from ref 5
%and then interpolating it
vL1=@(T)(0.1436.*T+65.991); 
vL2=@(T)(0.2057*T+86.504); 
vL3=@(T)(0.1389.*T+65.106);

%Defined the Lambda stuff for equation 6

L11=@(T)(1);
L12=@(T)((vL2(T)/vL1(T))*exp(-(866.47)/(R*T)));
L13=@(T)((vL3(T)/vL1(T))*exp(-(-416.33)/(R*T)));
L21=@(T)((vL1(T)/vL2(T))*exp(-(-629.14)/(R*T)));
L22=@(T)(1);
L23=@(T)((vL3(T)/vL2(T))*exp(-(-45.06)/(R*T)));
L31=@(T)((vL1(T)/vL3(T))*exp(-(911.48)/(R*T)));
L32=@(T)((vL2(T)/vL3(T))*exp(-(264.44)/(R*T)));
L33=@(T)(1);

%Define activity coefficient equation 5
%activity coefficients (g) equation 5
g1=@(x,T)(exp(1-log(((x(1)*L11(T))+(x(2)*L12(T))+(x(3)*L13(T))))-(((x(1)*L11(T))/((x(1)*L11(T))+(x(2)*L12(T))+(x(3)*L13(T))))+((x(2)*L21(T))/((x(1)*L21(T))+(x(2)*L22(T))+(x(3)*L23(T))))+((x(3)*L31(T))/((x(1)*L31(T))+(x(2)*L32(T))+(x(3)*L33(T)))))));
g2=@(x,T)(exp(1-log(((x(1)*L21(T))+(x(2)*L22(T))+(x(3)*L23(T))))-(((x(1)*L12(T))/((x(1)*L11(T))+(x(2)*L12(T))+(x(3)*L13(T))))+((x(2)*L22(T))/((x(1)*L21(T))+(x(2)*L22(T))+(x(3)*L23(T))))+((x(3)*L32(T))/((x(1)*L31(T))+(x(2)*L32(T))+(x(3)*L33(T)))))));
g3=@(x,T)(exp(1-log(((x(1)*L31(T))+(x(2)*L32(T))+(x(3)*L33(T))))-(((x(1)*L13(T))/((x(1)*L11(T))+(x(2)*L12(T))+(x(3)*L13(T))))+((x(2)*L23(T))/((x(1)*L21(T))+(x(2)*L22(T))+(x(3)*L23(T))))+((x(3)*L33(T))/((x(1)*L31(T))+(x(2)*L32(T))+(x(3)*L33(T)))))));

%Define Antoine equation
P1sat=@(T)((10^(6.84498-(1203.526/((T-273.15)+222.863))))*0.001315789);
P2sat=@(T)((10^(6.90240-(1268.115/((T-273.15)+216.9))))*0.001315789);
P3sat=@(T)((10^(6.95334-(1343.943/((T-273.15)+219.377))))*0.001315789);

[Star Strider]

How does:

yD=[0.3291 0.529 0.1418];

relate to:

x2*gamma2*P2sat=y2*1
x3*gamma3*P3sat=y3*1
x1*gamma1*P1sat+x2*gamma2*P2sat+x3*gamma3*P3sat=1
x1+x2+x3=1

I can’t make sense of your code (and I’m a few decades past my undergraduate PChem course).