Your belief has preciously little value. Faith in mathematics tends to be a waste of mental energy. Let me expand your equations, looking at them one at a time.

0 < beta_bar

beta_bar < 1

0 < sigma_bar

0 < phi_bar

0 < phi_pi_bar

0 < phi_y_bar

So those are simply lower and upper bounds on the problem. Solve does not understand inequalities, because they never give a unique solution.

syms g_23 phi_y_bar phi_pi_bar theta_bar phi_bar sigma_bar beta_bar

g_23 = solve(g_23^2 - 6835969944064461/9007199254740992 == 0)

g_23 =

That gave us two potential solutions for g_23. One is positive, one negative, but you never said anything about what g_23 should be.

phi_y_bar = solve(phi_y_bar - 3907/20000 == 0)

phi_y_bar =

phi_pi_bar = solve(phi_pi_bar - 14709/10000 == 0)

phi_pi_bar =

Those pair of equations simply assign values to variables. Why would you put lower and upper bounds on something when you then assign it a value?

Next, we see the equation for theta_bar. It has a trivial solution, regardless of the value of g_23.

theta_bar = solve(g_23*theta_bar - (7846842892884325*theta_bar)/9007199254740992 == 0,theta_bar)

Yep. theta_bar is just ZERO. You can see that, if you factor out theta_bar.

Next, we see an equation for phi_bar. I'll tell vpa to just write out the equation itself. Remember, that you told us that phi_bar MUST be positive.

vpa((11137*g_23)/10000 - (7846842892884325*phi_bar)/9007199254740992 + g_23*phi_bar - 1092378616225659/1125899906842624 == 0,5)

ans =

Do you see the first case? That tells us that phi_bar is NEGATIVE. Or the second term offers the possibility that phi_bar is EXACTLY zero. But you told us that phi_bar MUST be positive. Neither of the solutions are valid.

The fact is, we can stop right now. There are NO solutions to your problem.

While I could look at the rest of your problem, there is no need to proceed further. Well, I could write them down. This next one again tells us that phi_bar is NEGATIVE. But since we know it is not allowed to be negative...

(6250*phi_bar)/6197 - (2659174770252075*theta_bar)/1125899906842624 - (12447*phi_bar*theta_bar)/6197 + phi_bar*theta_bar^2 + (11137*theta_bar^2)/10000 + 55685/49576 == 0

ans =

sigma_bar - 11137/10000 == 0

ans =

beta_bar*theta_bar - (6197*theta_bar)/6250 == 0

ans = (11137*beta_bar)/10000 - (6197*phi_bar)/6250 + beta_bar*phi_bar - 1243281529372025/1125899906842624 == 0

ans =

Again, faith is a waste of mental energy. This is simply not a precision problem, but a question of existence at all.