How do I use SymPy to simplify by substitution a number of equations to get to a single simplified equation?

Question

My question is exactly like the title of my post. Attached is an image with an example of what I would like to achieve.

Description of problem I have updated this file after the original post and after the problem was solved to correct a typo.

Note Vref in the picture of the description is zero.

The attempts to get the SymPy functions mentioned to work failed.

Answer 1

Finding the solution you are looking for is really easy. First, you find Vx from eq1, substitutes it into eq2 and solve it for Vo:

from sympy import *
Vi, Vo, Vx = symbols("Vi, Vo, Vx")
R1, R2, R3, R4 = symbols("R1:5")

eq1 = Eq((0 - Vi) / R1, (Vi - Vx) / R2)
eq2 = Eq((Vi - Vx) / R2, Vx / R4 + (Vx - Vo) / R3)
Vx_expr = solve(eq1, Vx)[0]
Vo_expr = solve(eq2.subs(Vx, Vx_expr), Vo)[0]
Vo_expr

Now, if you want to transform it into the output you see on your screenshot, you have to use expression manipulation, which is pure masochism for a SymPy's novice. I'll show you how I'd do it.

In the following, you'll see several sympy's function in action. It is your job to read their documentation.

Vo_expr = Vo_expr.expand().collect(Vi)
Vo_expr

Now, I want to see the order of the arguments of the big addition:

Vo_expr.args[1].args

I select the terms that I need to modify:

term1 = Add(*[t for i, t in enumerate(Vo_expr.args[1].args) if i in [3, 4]])
display(term1, term1.together())

term2 = Add(*[t for i, t in enumerate(Vo_expr.args[1].args) if i in [1, 2]])
display(term2, term2.together())

Then, I substitute the terms with their modified forms:

Vo_final = Vo_expr.subs({term1: term1.together(), term2: term2.together()})
Vo_final

Finally, I verify that the result is equal to the original expression:

Vo_final.equals(Vo_expr)
# out: True