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Solving simultaneous equations with sympy

This document is a tutorial for how to use the Python module sympy to solve simultaneous equations. Since sympy does this so well, there is no need to implement it within reliability, but users may find this tutorial helpful as problems involving physics of failure will often require the solution of simultaneous equations. sympy is not installed by default when you install reliability so users following this tutorial will need to ensure sympy is installed on their machine. The following three examples should be sufficient to illustrate how to use sympy for solving simultaneous equations. Further examples are available in the sympy documentation.

Example 1

\(\text{Eqn 1:} \hspace{11mm} x + y = 5\)

\(\text{Eqn 2:} \hspace{11mm} x^2 + y^2 = 17\)

Solving with sympy:

import sympy as sym
x,y = sym.symbols('x,y')
eq1 = sym.Eq(x+y,5)
eq2 = sym.Eq(x**2+y**2,17)
result = sym.solve([eq1,eq2],(x,y))
print(result)

'''
[(1, 4), (4, 1)] #these are the solutions for x,y. There are 2 solutions because the equations represent a line passing through a circle.
'''

Example 2

\(\text{Eqn 1:} \hspace{11mm} a1000000^b = 119.54907\)

\(\text{Eqn 2:} \hspace{11mm} a1000^b = 405\)

Solving with sympy:

import sympy as sym
a,b = sym.symbols('a,b')
eq1 = sym.Eq(a*1000000**b,119.54907)
eq2 = sym.Eq(a*1000**b,405)
result = sym.solve([eq1,eq2],(a,b))
print(result)

'''
[(1372.03074854535, -0.176636273742481)] #these are the solutions for a,b
'''

Example 3

\(\text{Eqn 1:} \hspace{11mm} 2x^2 +y + z = 1\)

\(\text{Eqn 2:} \hspace{11mm} x + 2y + z = c_1\)

\(\text{Eqn 3:} \hspace{11mm} -2x + y = -z\)

The actual solution to the above set of equations is:

\(\hspace{21mm} x = -\frac{1}{2}+\frac{\sqrt{3}}{2}\)

\(\hspace{21mm} y = c_1 - \frac{3\sqrt{3}}{2}+\frac{3}{2}\)

\(\hspace{21mm} z = -c_1 - \frac{5}{2}+\frac{5\sqrt{3}}{2}\)

and a second solution:

\(\hspace{21mm} x = -\frac{1}{2}-\frac{\sqrt{3}}{2}\)

\(\hspace{21mm} y = c_1 + \frac{3\sqrt{3}}{2}+\frac{3}{2}\)

\(\hspace{21mm} z = -c_1 - \frac{5}{2}-\frac{5\sqrt{3}}{2}\)

Solving with sympy:

import sympy as sym
x,y,z = sym.symbols('x,y,z')
c1 = sym.Symbol('c1')
eq1 = sym.Eq(2*x**2+y+z,1)
eq2 = sym.Eq(x+2*y+z,c1)
eq3 = sym.Eq(-2*x+y,-z)
result = sym.solve([eq1,eq2,eq3],(x,y,z))
print(result)

'''
[(-1/2 + sqrt(3)/2, c1 - 3*sqrt(3)/2 + 3/2, -c1 - 5/2 + 5*sqrt(3)/2), (-sqrt(3)/2 - 1/2, c1 + 3/2 + 3*sqrt(3)/2, -c1 - 5*sqrt(3)/2 - 5/2)]
'''

Note

If you are using an iPython notebook, the display abilities are much better than the command line interface, so you can simply add sym.init_printing() after the import line and your equations should be displayed nicely.

A special thanks to Brigham Young University for offering this tutorial.