sympy

Use SymPy's solveset for algebraic equations, linsolve for linear systems, nonlinsolve for nonlinear systems, and dsolve for differential equations. Define symbols first with symbols(), then pass the equation and variable, for example solveset(x**2 - 4, x) returns {-2, 2}.

Use SymPy when you need exact symbolic results like sqrt(2), symbolic derivatives, integrals, or equation solving. Use NumPy when floating-point approximations are sufficient and you need fast numerical computation over large arrays.

Use lambdify to convert a symbolic expression into a NumPy-compatible function, for example f = lambdify(x, expr, 'numpy'). This is far faster than calling subs() and evalf() repeatedly in a loop.

Writing 0.5 * x creates a floating-point value instead of an exact rational. Use Rational(1, 2) or S(1)/2 to keep arithmetic exact, and call evalf() only when you need a numerical approximation.

Yes. The latex() function converts expressions to LaTeX for documents, and sympy.utilities.codegen generates C or Fortran code. The autowrap function can even compile expressions into callable Python functions.

Simplification often depends on symbol assumptions. Define symbols with constraints like positive=True or real=True so functions like sqrt(x**2) simplify correctly, and try specific functions like trigsimp, factor, or expand instead of the general simplify.