Solve an Equation Algebraically

Use SymPy to solve an equation algebraically (symbolically). For example, solving \(x^2 = y\) for \(x\) yields \(x \in \{-\sqrt{y},\sqrt{y}\}\).

Alternatives to consider

Solving Functions

There are two high-level functions to solve equations, solve() and solveset(). Here is an example of each:

solve()

>>> from sympy.abc import x, y
>>> from sympy import solve
>>> solve(x**2 - y, x, dict=True)
[{x: -sqrt(y)}, {x: sqrt(y)}]

solveset()

>>> from sympy import solveset
>>> from sympy.abc import x, y
>>> solveset(x**2 - y, x)
{-sqrt(y), sqrt(y)}

Here are recommendations on when to use:

  • solve()

    • You want to get explicit symbolic representations of the different values a variable could take that would satisfy the equation.

    • You want to substitute those explicit solution values into other equations or expressions involving the same variable using subs()

  • solveset()

    • You want to represent the solutions in a mathematically precise way, using mathematical sets.

    • You want a representation of all the solutions, including if there are infinitely many.

    • You want a consistent input interface.

    • You want to limit the domain of the solutions to any arbitrary set.

    • You do not need to programmatically extract solutions from the solution set: solution sets cannot necessarily be interrogated programmatically.

Guidance

Refer to Include the Variable to be Solved for in the Function Call and Ensure Consistent Formatting From solve().

Solve an Equation Algebraically

You can solve an equation in several ways. The examples below demonstrate using both solve() and solveset() where applicable. You can choose the function best suited to your equation.

Make Your Equation Into an Expression That Equals Zero

Use the fact that any expression not in an Eq (equation) is automatically assumed to equal zero (0) by the solving functions. You can rearrange the equation \(x^2 = y\) to \(x^2 - y = 0\), and solve that expression. This approach is convenient if you are interactively solving an expression which already equals zero, or an equation that you do not mind rearranging to \(expression = 0\).

>>> from sympy import solve, solveset
>>> from sympy.abc import x, y
>>> solve(x**2 - y, x, dict=True)
[{x: -sqrt(y)}, {x: sqrt(y)}]
>>> solveset(x**2 - y, x)
{-sqrt(y), sqrt(y)}

Put Your Equation Into Eq Form

Put your equation into Eq form, then solve the Eq. This approach is convenient if you are interactively solving an equation which you already have in the form of an equation, or which you think of as an equality. It also helps to prevent sign errors when subtracting one side from the other.

>>> from sympy import Eq, solve, solveset
>>> from sympy.abc import x, y
>>> eqn = Eq(x**2, y)
>>> eqn
Eq(x**2, y)
>>> solutions = solve(eqn, x, dict=True)
>>> print(solutions)
[{x: -sqrt(y)}, {x: sqrt(y)}]
>>> solutions_set = solveset(eqn, x)
>>> print(solutions_set)
{-sqrt(y), sqrt(y)}
>>> for solution_set in solutions_set:
...     print(solution_set)
sqrt(y)
-sqrt(y)

Understand How Eq Is Interpreted

An Equality such as Eq(lhs, rhs) is a symbolic Boolean relation: it represents the logical statement that the two sides are equal. Operations on the relation itself, such as simplify(), must preserve that truth-value meaning.

Some equation-oriented APIs also accept Eq(lhs, rhs) as convenient syntax for the formal equation lhs - rhs = 0. The interpretation therefore depends on the API consuming the Eq:

Context

Treatment of Eq(lhs, rhs)

simplify(Eq(...)) and other operations on the relation itself

A logical relation; transformations must preserve its truth value.

solve(Eq(...), x)

Equation input; it is converted to lhs - rhs = 0 before solving.

solveset(Eq(...), x)

Equation input while the relation depends on x; a relation independent of x is evaluated as a logical condition.

Poly(Eq(...)) and linear_eq_to_matrix(Eq(...))

Equation input representing lhs - rhs = 0.

Ordered relations such as <, <=, >, and >= passed to solvers

Relations whose ordering semantics must be preserved.

This equation-oriented interpretation does not change the semantics of the Equality as a symbolic relation. Consider the following:

>>> solve(Eq(x - y, 0), x)
[y]

The equation solver has interpreted this as asking for solutions of the formal equation x - y = 0, but this does not mean that Eq(x - y, 0) and Eq(x, y) must have the same truth value for non-finite values:

>>> from sympy import oo
>>> Eq(oo, oo)
True
>>> Eq(oo - oo, 0)
False

An Eq can evaluate to True or False when it is constructed, before an equation-oriented API receives it. When an API needs an unevaluated Equality as equation syntax, Eq(..., evaluate=False) can be used explicitly.

Restrict the Domain of Solutions

By default, SymPy will return solutions in the complex domain, which also includes purely real and imaginary values. Here, the first two solutions are real, and the last two are imaginary:

>>> from sympy import Symbol, solve, solveset
>>> x = Symbol('x')
>>> solve(x**4 - 256, x, dict=True)
[{x: -4}, {x: 4}, {x: -4*I}, {x: 4*I}]
>>> solveset(x**4 - 256, x)
{-4, 4, -4*I, 4*I}

To restrict returned solutions to real numbers, or another domain or range, the different solving functions use different methods.

For solve(), place an assumption on the symbol to be solved for, \(x\)

>>> from sympy import Symbol, solve
>>> x = Symbol('x', real=True)
>>> solve(x**4 - 256, x, dict=True)
[{x: -4}, {x: 4}]

or restrict the solutions with standard Python techniques for filtering a list such as a list comprehension:

>>> from sympy import Or, Symbol, solve
>>> x = Symbol('x', real=True)
>>> expr = (x-4)*(x-3)*(x-2)*(x-1)
>>> solution = solve(expr, x)
>>> print(solution)
[1, 2, 3, 4]
>>> solution_outside_2_3 = [v for v in solution if (v.is_real and Or(v<2,v>3))]
>>> print(solution_outside_2_3)
[1, 4]

For solveset(), limit the output domain in the function call by setting a domain

>>> from sympy import S, solveset
>>> from sympy.abc import x
>>> solveset(x**4 - 256, x, domain=S.Reals)
{-4, 4}

or by restricting returned solutions to any arbitrary set, including an interval:

>>> from sympy import Interval, pi, sin, solveset
>>> from sympy.abc import x
>>> solveset(sin(x), x, Interval(-pi, pi))
{0, -pi, pi}

and if you restrict the solutions to a domain in which there are no solutions, solveset() will return the empty set, EmptySet:

>>> from sympy import solveset, S
>>> from sympy.abc import x
>>> solveset(x**2 + 1, x, domain=S.Reals)
EmptySet

Explicitly Represent Infinite Sets of Possible Solutions

solveset() can represent infinite sets of possible solutions and express them in standard mathematical notation, for example \(\sin(x) = 0\) for \(x = n * \pi\) for every integer value of \(n\):

>>> from sympy import pprint, sin, solveset
>>> from sympy.abc import x
>>> solution = solveset(sin(x), x)
>>> pprint(solution)
{2*n*pi | n in Integers} U {2*n*pi + pi | n in Integers}

However, solve() will return only a finite number of solutions:

>>> from sympy import sin, solve
>>> from sympy.calculus.util import periodicity
>>> from sympy.abc import x
>>> f = sin(x)
>>> solve(f, x)
[0, pi]
>>> periodicity(f, x)
2*pi

solve() tries to return just enough solutions so that all (infinitely many) solutions can generated from the returned solutions by adding integer multiples of the periodicity() of the equation, here \(2\pi\).

Use the Solution Result

Substitute Solutions From solve() Into an Expression

You can substitute solutions from solve() into an expression.

A common use case is finding the critical points and values for a function \(f\). At the critical points, the Derivative equals zero (or is undefined). You can then obtain the function values at those critical points by substituting the critical points back into the function using subs(). You can also tell if the critical point is a maxima or minima by substituting the values into the expression for the second derivative: a negative value indicates a maximum, and a positive value indicates a minimum.

>>> from sympy.abc import x
>>> from sympy import solve, diff
>>> f = x**3 + x**2 - x
>>> derivative = diff(f, x)
>>> critical_points = solve(derivative, x, dict=True)
>>> print(critical_points)
[{x: -1}, {x: 1/3}]
>>> point1, point2 = critical_points
>>> print(f.subs(point1))
1
>>> print(f.subs(point2))
-5/27
>>> curvature = diff(f, x, 2)
>>> print(curvature.subs(point1))
-4
>>> print(curvature.subs(point2))
4

solveset() Solution Sets Cannot Necessarily Be Interrogated Programmatically

If solveset() returns a finite set (class FiniteSet), you can iterate through the solutions:

>>> from sympy import solveset
>>> from sympy.abc import x, y
>>> solution_set = solveset(x**2 - y, x)
>>> print(solution_set)
{-sqrt(y), sqrt(y)}
>>> solution_list = list(solution_set)
>>> print(solution_list)
[sqrt(y), -sqrt(y)]

However, for more complex results, it may not be possible to list the solutions:

>>> from sympy import S, solveset, symbols
>>> x, y = symbols('x, y')
>>> solution_set = solveset(x**2 - y, x, domain=S.Reals)
>>> print(solution_set)
Intersection({-sqrt(y), sqrt(y)}, Reals)
>>> list(solution_set)
Traceback (most recent call last):
    ...
TypeError: The computation had not completed because of the undecidable set
membership is found in every candidates.

In this case, it is because, if \(y\) is negative, its square root would be imaginary rather than real and therefore outside the declared domain of the solution set. By declaring \(y\) to be real and positive, SymPy can determine that its square root is real, and thus resolve the intersection between the solutions and the set of real numbers:

>>> from sympy import S, Symbol, solveset
>>> x = Symbol('x')
>>> y = Symbol('y', real=True, positive=True)
>>> solution_set = solveset(x**2 - y, x, domain=S.Reals)
>>> print(solution_set)
{-sqrt(y), sqrt(y)}
>>> list(solution_set)
[sqrt(y), -sqrt(y)]

Alternatively, you can extract the sets from the solution set using args, then create a list from the set containing the symbolic solutions:

>>> from sympy import S, solveset, symbols
>>> x, y = symbols('x, y')
>>> solution_set = solveset(x**2 - y, x, domain=S.Reals)
>>> print(solution_set)
Intersection({-sqrt(y), sqrt(y)}, Reals)
>>> solution_set_args = solution_set.args
>>> print(solution_set.args)
(Reals, {-sqrt(y), sqrt(y)})
>>> list(solution_set_args[1])
[sqrt(y), -sqrt(y)]

Options That Can Speed up solve()

Refer to solving guidance.

Not All Equations Can Be Solved

Equations With No Closed-Form Solution

Some equations have no closed-form solution, in which case SymPy may return an empty set or give an error. For example, the following transcendental equation has no closed-form solution:

>>> from sympy import cos, solve
>>> from sympy.abc import x
>>> solve(cos(x) - x, x, dict=True)
Traceback (most recent call last):
    ...
NotImplementedError: multiple generators [x, cos(x)]
No algorithms are implemented to solve equation -x + cos(x)

Equations Which Have a Closed-Form Solution, and SymPy Cannot Solve

It is also possible that there is an algebraic solution to your equation, and SymPy has not implemented an appropriate algorithm. If that happens, or SymPy returns an empty set or list when there is a mathematical solution (indicating a bug in SymPy), please post it on the mailing list, or open an issue on SymPy’s GitHub page. Until the issue is resolved, you can solve your equation numerically instead.

Report a Bug

If you find a bug with a solving function, please post the problem on the SymPy mailing list. Until the issue is resolved, you can use a different method listed in Alternatives to consider.