The following code shows the two equations, here in variables A, B and constants alpha, beta, are linear transforms of same as captured in: h(alpha*A,beta) = g(alpha*A,B), h(A,beta) = g(A,B). Relations are stored in EQ1s, EQf and EQ1.

restart;
eq1:=(1-Pi*x/2)*(f/2/E)*yc-1/3=Int(t^4/sqrt((t^2-(f/2/E)*yc)^2+ya^2),t=0..1);
eq2:=x^2*yc*(1-Pi*x/2)-1/3=Int(t^4/sqrt((t^2-x^2*yc)^2+ya^2),t=0..1);

eq2:=subs(t^4-2*t^2*x^2*yc+x^4*yc^2+ya^2 = (t^2-x^2*yc)^2+ya^2,eq2);

EQ1s:=A = 1/2*yc/E*f, B = ya^2, alpha = 2*E*x^2/f, beta = x;
G:=(q,r)->Int(t^4/((t^2-q)^2+r)^(1/2),t = 0 .. 1);;
H:=(q,r)->q*(1-1/2*Pi*r)-1/3;
EQf:=h=H,g=G;
EQ1:=h(alpha*A,beta) = g(alpha*A,B), h(A,beta) = g(A,B);

#show
g=eval(G);h=eval(H);
EQf;
EQ1s;
EQ1;
form:=(eval@subs)([EQf],[EQ1s],[EQ1]);
test:=map(evalb,form-[eq2,eq1]);
(eval@subs)(EQf,[EQ1]);
EQ1;

The matrix M can be inverted through the use of diagonalization. The inverse matrix entries are very large. Direct computation of the inverse fails with object too large error. A proposition of computation path follows. Observed sturcture of M matrix is posted in M_Qver (using variables Q1..Q7) and observed sturcture of M inverse matrix is posted in Mi_Pver (using variables P1..P12). You can solve for the new variables.
Use equation
M = M_Qver  
to get Q variables from a,b,c,d,e,w. Convert
0 = M_Qver &* Mi_Pver - 1.
0 = Mi_Pver &* M_Qver - 1
into a system of linears in variables P1..P12. to get P variables from Q variables. Solve for P variables and substitute into Mi_Pver the inverse of M. In following code matrix PMi last column contains values of P1..12. This computes in minutes.

Code:

restart:
with(linalg):

M:=matrix([ [a+b+c+d+e,a+b+d+w*e,a+b+d+w^2*e,a+b+c,a+b,a+b,a+c,a, a,a+c,a,a], [a+b+d+w*e,a+b+c+d+e,a+b+d+w*e,a+b,a+b+c,a+b,a,a+c,a, a,a+c, a], [a+b+d+w^2*e,a+b+d+w*e,a+b+c+d+e,a+b,a+b,a+b+c,a,a,a+c,a, a,a+c], [a+b+c,a+b,a+b,a+b+c+d+e,a+b+d+w*e,a+b+d+w^2*e,a+c,a,a, a+c,a, a], [a+b,a+b+c,a+b,a+b+d+w*e,a+b+c+d+e,a+b+d+w*e,a,a+c,a,a, a+c, a], [a+b,a+b,a+b+c, a+b+d+w^2*e,a+b+d+w*e, a+b+c+d+e,a,a,a+c, a, a,a+c], [a+c, a,a,a+c,a, a,a+b+c+d+e,a+b+d+w*e,a+b+d+w^2*e, a+b+c, a+b,a+b], [a,a+c,a,a, a+c,a,a+b+d+w*e,a+b+c+d+e,a+b+d+w*e, a+b, a+b+c, a+b], [a,a, a+c,a, a, a+c, a+b+d+w^2*e,a+b+d+w*e,a+b+c+d+e, a+b,a+b,a+b+c], [a+c,a,a,a+c,a,a,a+b+c, a+b,a+b,a+b+c+d+e, a+b+d+w*e, a+b+d+w^2*e], [a, a+c,a,a,a+c,a,a+b,a+b+c, a+b, a+b+d+w*e, a+b+c+d+e, a+b+d+w*e], [a,a,a+c,a,a,a+c,a+b, a+b,a+b+c,a+b+d+w^2*e,a+b+d+w*e, a+b+c+d+e]]);

M_subs:={a = Q3, a+b = Q6, a+b+c = Q7, a+b+d+w^2*e = Q2, a+b+d+w*e = Q5, a+b+c+d+e = Q1, a+c = Q4};
Q_vals := {Q1 = a+b+c+d+e, Q5 = a+b+d+w*e, Q7 = a+b+c, Q2 = a+b+d+w^2*e, Q4 = a+c, Q3 = a, Q6 = a+b};
Q_ident:={Q3 = Q4-Q7+Q6};

M_Qver := matrix([[Q1, Q5, Q2, Q7, Q6, Q6, Q4, Q3, Q3, Q4, Q3, Q3], [Q5, Q1, Q5, Q6, Q7, Q6, Q3, Q4, Q3, Q3, Q4, Q3], [Q2, Q5, Q1, Q6, Q6, Q7, Q3, Q3, Q4, Q3, Q3, Q4], [Q7, Q6, Q6, Q1, Q5, Q2, Q4, Q3, Q3, Q4, Q3, Q3], [Q6, Q7, Q6, Q5, Q1, Q5, Q3, Q4, Q3, Q3, Q4, Q3], [Q6, Q6, Q7, Q2, Q5, Q1, Q3, Q3, Q4, Q3, Q3, Q4], [Q4, Q3, Q3, Q4, Q3, Q3, Q1, Q5, Q2, Q7, Q6, Q6], [Q3, Q4, Q3, Q3, Q4, Q3, Q5, Q1, Q5, Q6, Q7, Q6], [Q3, Q3, Q4, Q3, Q3, Q4, Q2, Q5, Q1, Q6, Q6, Q7], [Q4, Q3, Q3, Q4, Q3, Q3, Q7, Q6, Q6, Q1, Q5, Q2], [Q3, Q4, Q3, Q3, Q4, Q3, Q6, Q7, Q6, Q5, Q1, Q5], [Q3, Q3, Q4, Q3, Q3, Q4, Q6, Q6, Q7, Q2, Q5, Q1]]);

Mi_Pver := matrix([[P10, P9, P1, P2, P5, P3, P6, P7, P8, P6, P7, P8], [P9, P11, P9, P5, P12, P5, P7, P4, P7, P7, P4, P7], [P1, P9, P10, P3, P5, P2, P8, P7, P6, P8, P7, P6], [P2, P5, P3, P10, P9, P1, P6, P7, P8, P6, P7, P8], [P5, P12, P5, P9, P11, P9, P7, P4, P7, P7, P4, P7], [P3, P5, P2, P1, P9, P10, P8, P7, P6, P8, P7, P6], [P6, P7, P8, P6, P7, P8, P10, P9, P1, P2, P5, P3], [P7, P4, P7, P7, P4, P7, P9, P11, P9, P5, P12, P5], [P8, P7, P6, P8, P7, P6, P1, P9, P10, P3, P5, P2], [P6, P7, P8, P6, P7, P8, P2, P5, P3, P10, P9, P1], [P7, P4, P7, P7, P4, P7, P5, P12, P5, P9, P11, P9], [P8, P7, P6, P8, P7, P6, P3, P5, P2, P1, P9, P10]]);

P_vector:=[P1, P2, P3, P4, P5, P6, P7, P8, P9, P10, P11, P12];
QP_set:=convert(evalm(M_Qver &* Mi_Pver - 1),set) union convert(evalm(Mi_Pver &* M_Qver - 1),set);
QP_set:=subs(Q_ident,QP_set):

PM:=genmatrix(QP_set,P_vector,true);
PMi:=gaussjord(PM,coldim(PM)-1):

Expecting real values over the real range you'd choose this function:

piecewise(0 < x,2*(-1+x)/x^(5/3),x = 0,FAIL,x < 0,(-1+I*3^(1/2))*(-1+x)/x^(5/3))

note this function is real valued but floating point valuation produces an imaginary component, so use Re.

plot(Re(piecewise(0 < x,2*(-1+x)/x^(5/3),x = 0,FAIL,x < 0,(-1+I*3^(1/2))*(-1+x)/x^(5/3))),x=-4..4,view=[-4..4,-4..4]);

or

plot(piecewise(0 < x,2*(-1+x)/x^(5/3),x = 0,FAIL,x < 0,-2*(-1+x)/(-x)^(5/3)),x=-4..4,view=[-4..4,-4..4]);

This was written with an old version of Maple so I apologize for any difficulties.

restart;
h:=x->-2/x^(5/3)+2/x^(2/3);
assume(x,complex,q,complex);
h(x);
rationalize(h(x));
E:=(denom@rationalize)(1/(q-rationalize(h(x))));
sol:=[solve](E,q);

atry:=map(x->piecewise(Im(x)=0,Re(x),FAIL),sol):
ans:=unapply(eval(atry,1),x)(-1/2);
ans:=remove(type,ans,map(identical,{FAIL}));
ans[1];
evalf(ans[1],20);

                                   2/3
                                6 2


                        9.5244063118091968488