# How to solve symbolic equations

## Description

Once we’ve expressed an equation or system of equations using the technique from how to write symbolic equations, we often want the software to solve the equation or system of equations for us.

Related tasks:

## Using SymPy, in Python

This answer assumes you have imported SymPy as follows.

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from sympy import * # load all math functions
init_printing( use_latex='mathjax' ) # use pretty math output

If your equation has just one variable, simply call `solve`

on it.
Note that you may get a list of more than one solution.

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var( 'x' )
equation = Eq( x**2 + 3*x, -x + 9 )
solve( equation )

$\displaystyle \left[ -2 + \sqrt{13}, \ - \sqrt{13} - 2\right]$

Sometimes you get no solutions, which is shown as a Python empty list.

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solve( Eq( x+1, x+2 ) )

$\displaystyle \left[ \right]$

Sometimes the answers include complex numbers.

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solve( Eq( x**3, -1 ) )

$\displaystyle \left[ -1, \ \frac{1}{2} - \frac{\sqrt{3} i}{2}, \ \frac{1}{2} + \frac{\sqrt{3} i}{2}\right]$

To restrict the solution to the real numbers, use `solveset`

instead,
and specify the real numbers as the domain.

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solveset( Eq( x**3, -1 ), domain=S.Reals )

$\displaystyle \left\{-1\right\}$

(If `solveset`

gives no solution, it shows it as the empty set symbol, $\emptyset$.)

You can solve systems of equations by calling `solve`

on them.

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var( 'x y' )
system = [
Eq( x + 2*y, 1 ),
Eq( x - 9*y, 5 )
]
solve( system )

$\displaystyle \left\{ x : \frac{19}{11}, \ y : - \frac{4}{11}\right\}$

Content last modified on 24 July 2023.

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