How to substitute and simplify an expression in sympy?

Question

I want to substitute expression

(k*x_2 + m)/(x_2 + 1) + (k*x_1 + m)/(x_1 + 1)

with

x_1 + x_2 = -(2*k*m-8)/k**2
x_1 * x_2 = m**2/k**2

and simplify it, which should have following result:

8*(k + m)/(k**2 - 2*k*m + m**2 + 8)

I have tried .subs({x1+x2: blahblah, x1*x2: blahblah}). Indeed, it does substitute some x1+x2 and x1*x2 with blahblah, but it still remains some x1+x2 in the expression. How to solve this problem?

Thanks!

Answer 1

Maybe the definition of x1 and x2 had something wrong. I get the desired result using what was shown:

>>> a = (k*x_2 + m)/(x_2 + 1) + (k*x_1 + m)/(x_1 + 1)
>>> b = collect(collect(cancel(a), m), k)
>>> b.subs({x_1 + x_2: -(2*k*m - 8)/k**2, x_1*x_2: m**2/k**2}).simplify()
8*(k + m)/(k**2 - 2*k*m + m**2 + 8)

Another way to think of this problem is that it requires x_1 and x_2 to be eliminated from a given the relationships you define. So if we solve the coupled relationship for x_1 and x_2 and substitute those into a we will have the desired result:

>>> e2  # = Eq(x_1 + x_2 , (-2*k*m + 8)/k**2)
x_1 + x_2 == (-2*k*m + 8)/k**2
>>> e3
x_1*x_2 == m**2/k**2
>>> x1x2 = solve((e2,e3),x_1,x_2,dict=True)  # two solutions are given
>>> a.subs(x1x2[0]).simplify()  # use either solution; the result is the same
8*(k + m)/(k**2 - 2*k*m + m**2 + 8)

Answer 2

SymPy only substitute expressions for expressions if it can find them exactly (with a few little additions like matching 2*x against 4*x).

You'll need to rewrite the expression to have those terms like you want. cancel will put everything under one denominator, and collect will let you convert m*x_1 + m*x_2 into m*(x_1 + x_2) in that expression, so that you have x_1 + x_2. In short:

>>> a = (k*x_2 + m)/(x_2 + 1) + (k*x_1 + m)/(x_1 + 1)
>>> b = collect(collect(cancel(a), m), k)
>>> b
(k*(2*x_1*x_2 + x_1 + x_2) + m*(x_1 + x_2 + 2))/(x_1*x_2 + x_1 + x_2 + 1)
>>> b.subs({x1 + x2: -(2*k*m - 8)/k**2, x1*x2: m**2/k**2})
(k*(2*m**2/k**2 + (-2*k*m + 8)/k**2) + m*(x_1 + x_2 + 2))/(1 + m**2/k**2 + (-2*k*m + 8)/k**2)

This didn't seem to work completely, which I have opened a bug for https://github.com/sympy/sympy/issues/7475.