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SolveValues
✖
SolveValues

Updated in 14

✖
SolveValues[expr,vars]

gives the values of vars determined by the solutions of the system expr.

✖
SolveValues[expr,vars,dom]

uses solutions over the domain dom. Common choices of dom are Reals, Integers and Complexes.

Details and Options

  • The system expr can be any logical combination of:
  • lhs==rhsequations
    lhs!=rhsinequations
    lhs>rhs or lhs>=rhs inequalities
    expr∈domdomain specifications
    {x,y,…}∈regregion specification
    ForAll[x,cond,expr]universal quantifiers
    Exists[x,cond,expr]existential quantifiers
  • SolveValues[{expr1,expr2,…},vars] is equivalent to SolveValues[expr1&&expr2&&…,vars].
  • If a single variable is specified, the result is a list of values of the variable for which expr is True.
  • If a list of variables is specified, the result is a list of lists of values for the variables for which expr is True.
  • When a single variable is specified and a particular root of an equation has multiplicity greater than one, SolveValues gives several copies of the corresponding solution.
  • SolveValues[expr,vars] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
  • SolveValues[expr,vars,dom] restricts all variables and parameters to belong to the domain dom.
  • If dom is Reals or a subset such as Integers or Rationals, then all constants and function values are also restricted to be real.
  • SolveValues[expr&&vars∈Reals,vars,Complexes] solves for real values of variables, but function values are allowed to be complex.
  • SolveValues[expr,vars,Integers] solves Diophantine equations over the integers.
  • SolveValues[…,x∈reg,Reals] constrains x to be in the region reg. The different coordinates for x can be referred to using Indexed[x,i].
  • Algebraic variables in expr free of vars and of each other are treated as independent parameters.
  • SolveValues deals primarily with linear and polynomial equations.
  • When expr involves only polynomial equations and inequalities over real or complex domains, then SolveValues can always in principle solve directly for vars.
  • When expr involves transcendental conditions or integer domains, SolveValues will often introduce additional parameters in its results.
  • SolveValues can give explicit representations for solutions to all linear equations and inequalities over the integers and can solve a large fraction of Diophantine equations described in the literature.
  • When expr involves only polynomial conditions over real or complex domains, SolveValues[expr,vars] will always be able to eliminate quantifiers.
  • SolveValues gives generic solutions only. Solutions that are valid only when continuous parameters satisfy equations are removed. Other solutions that are only conditionally valid are expressed as ConditionalExpression objects.
  • Conditions included in ConditionalExpression solutions may involve inequalities, Element statements, equations and inequations on non-continuous parameters and equations with full-dimensional solutions. Inequations and NotElement conditions on continuous parameters and variables are dropped.
  • SolveValues may use non-equivalent transformations to find solutions of transcendental equations and hence it may not find some solutions and may not establish exact conditions on the validity of the solutions found. If this happens, an error message is issued.
  • SolveValues uses special efficient techniques for handling sparse systems of linear equations with approximate numerical coefficients.
  • The following options can be given:
  • Assumptions $Assumptionsassumptions on parameters
    Cubics Automaticwhether to use explicit radicals to solve all cubics
    GeneratedParameters Chow to name parameters that are generated
    InverseFunctions Automaticwhether to use symbolic inverse functions
    MaxExtraConditions 0how many extra equational conditions on continuous parameters to allow
    MaxRoots Infinitymaximum number of roots returned
    Method Automaticwhat method should be used
    Modulus 0modulus to assume for integers
    Quartics Automaticwhether to use explicit radicals to solve all quartics
    VerifySolutions Automaticwhether to verify solutions obtained using non-equivalent transformations
    WorkingPrecision Infinityprecision to be used in computations
  • With MaxExtraConditions->Automatic, only solutions that require the minimal number of equational conditions on continuous parameters are included.
  • With MaxExtraConditions->All, solutions that require arbitrary conditions on parameters are given and all conditions are included.
  • With MaxExtraConditions->k, only solutions that require at most k equational conditions on continuous parameters are included.
  • With Method->Reduce, SolveValues uses only equivalent transformations and finds all solutions.
  • SolveValues[eqns,…,Modulus->m] solves equations over the integers modulo m. With Modulus->Automatic, SolveValues will attempt to find the largest modulus for which the equations have solutions.

Examples

open allclose all

Basic Examples  (5)Summary of the most common use cases

Solve a quadratic equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-g05
Out[1]=1

Solve simultaneous equations in and :

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-bcw
Out[1]=1

Solve an equation over the reals:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-i1x0jj
Out[1]=1

Solve an equation over the positive integers:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-y3h7i
Out[1]=1

Solve equations in a geometric region:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-m1hbjg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-b209ra
Out[2]=2

Scope  (87)Survey of the scope of standard use cases

Basic Uses  (7)

Solutions are given as lists of values of the specified variables:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-sz7e7
Out[1]=1

Check that solutions satisfy the equations:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-pdsmq
Out[2]=2

If there are no solutions, SolveValues returns an empty list:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-en8k4e
Out[1]=1

Some of the variables may appear in the solutions as free parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-c4qd5v
Out[1]=1

Find solutions over specified domains:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-btcl0w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-cocpxs
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-fzldx9
Out[3]=3

Solve equations with coefficients involving a symbolic parameter:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-bw37uu
Out[1]=1

Plot the real parts of the solutions for y as a function of the parameter a:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-cfj46b
Out[2]=2

Solution of this equation over the reals requires conditions on the parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-c6oz7j
Out[1]=1

Use Normal to remove the conditions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-gmz5p9
Out[2]=2

Solution of this equation over the positive integers requires introduction of a new parameter:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-mcip9w
Out[1]=1

List the first 10 solutions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-dyk2qj
Out[2]=2

Complex Equations in One Variable  (16)

Polynomial equations solvable in radicals:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-b26xcn
Out[1]=1

To use general formulas for solving cubic equations, set CubicsTrue:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-c61ess
Out[2]=2

By default, SolveValues uses Root objects to represent solutions of general cubic equations:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-eisygm
Out[3]=3

General polynomial equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-l7qts7
Out[1]=1

Polynomial equations with multiple roots:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fg4dko
Out[1]=1

Find five roots of a polynomial of a high degree:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-jik1b0
Out[1]=1

Polynomial equations with symbolic coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-o9kfg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-ig0hgm
Out[2]=2

Algebraic equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-5itph
Out[1]=1

Complete solutions to transcendental equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-ei0sno
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-do4ma9
Out[2]=2

Partial solutions to transcendental equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-iec8fz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-euvlxi
Out[2]=2

SolveValues cannot find all solutions here:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-eo6zgh
Out[1]=1

Find three solutions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-oqfdgq
Out[2]=2

Univariate elementary function equations over bounded regions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-vopz12
Out[1]=1

Univariate holomorphic function equations over bounded regions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fssx58
Out[1]=1

Here SolveValues finds some solutions but is not able to prove there are no other solutions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-dlszza
Out[1]=1

Equation with a purely imaginary period over a vertical stripe in the complex plane:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-mlngs5
Out[1]=1

Find a specified number of roots of an unrestricted complex equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-bcvf0m
Out[1]=1

Symbolic functions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-bc3wia
Out[1]=1

Nonanalytic complex equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-chwdz4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-jqotg3
Out[2]=2

Systems of Complex Equations in Several Variables  (12)

Systems of linear equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-ed9vjf
Out[1]=1

Linear equations with symbolic coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-6r4vz
Out[1]=1

Underdetermined systems of linear equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cbonrc
Out[1]=1

Linear equations with no solutions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-b0z54x
Out[1]=1

Systems of polynomial equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-el7e2z
Out[1]=1

Find five out of a trillion roots of a polynomial system:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-jddb0
Out[1]=1

Polynomial equations with symbolic coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-bqycub
Out[1]=1

Algebraic equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-dbz6tx
Out[1]=1

Transcendental equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-innk5j
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-buzvxz
Out[2]=2

Find a specified number of solutions of transcendental equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-wl01fb
Out[1]=1

Square analytic systems over bounded boxes:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-83w7nf
Out[1]=1

Nonanalytic equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cega6t
Out[1]=1

Real Equations in One Variable  (13)

Polynomial equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-dhq5bv
Out[1]=1

Polynomial equations with multiple roots:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-dc26xt
Out[1]=1

Polynomial equations with symbolic coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-w7yylm
Out[1]=1

Algebraic equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-i4j3df
Out[1]=1

Piecewise equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-dyni8
Out[1]=1

Transcendental equations, solvable using inverse functions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-tfnsh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-obvohx
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-2jwjtn
Out[3]=3

Transcendental equations, solvable using special function zeros:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-hb5v2w
Out[1]=1

Transcendental inequalities, solvable using special function zeros:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-bb404h
Out[1]=1

Exp-log equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-dhdgkq
Out[1]=1

High-degree sparse polynomial equations:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-imbp1a
Out[2]=2

Algebraic equations involving high-degree radicals:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-penng
Out[3]=3

Equations involving non-rational real powers:

In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-i46va
Out[4]=4

Equation with a double root:

In[5]:=5
✖
https://wolfram.com/xid/0bdou600i-ppetdc
Out[5]=5

Tame elementary function equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-b65h95
Out[1]=1

Elementary function equations in bounded intervals:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-beor2b
Out[1]=1

Holomorphic function equations in bounded intervals:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-k3cxtw
Out[1]=1

Periodic elementary function equations over the reals:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-d6vn1
Out[1]=1

Systems of Real Equations and Inequalities in Several Variables  (10)

Linear systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fskac
Out[1]=1

Polynomial systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-mt2k2q
Out[1]=1

Quantified polynomial systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-oqx5qn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-d6xrhv
Out[2]=2

Algebraic systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cdubtk
Out[1]=1

Piecewise systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fphl0y
Out[1]=1

Transcendental systems, solvable using inverse functions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-s4nr6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-pu71op
Out[2]=2

Systems exp-log in the first variable and polynomial in the other variables:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-k1k6ul
Out[1]=1

Quantified system:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-egms13
Out[2]=2

Systems elementary and bounded in the first variable and polynomial in the other variables:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fvz3v9
Out[1]=1

Quantified system:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-ejllf0
Out[2]=2

Systems holomorphic and bounded in the first variable and polynomial in the other variables:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fb6mv5
Out[1]=1

Quantified system:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-hn9wkc
Out[2]=2

Square systems of analytic equations over bounded regions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-pfhd6f
Out[1]=1

Diophantine Equations  (11)

Linear systems of equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-sz2ju
Out[1]=1

Linear systems of equations and inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fulreq
Out[1]=1

Univariate polynomial equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-km7sh0
Out[1]=1

Binary quadratic equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-g1rl8c
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-bk2dj1
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-cwbm4i
Out[3]=3

Thue equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-enjls9
Out[1]=1

Sum of squares equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-hsrd0b
Out[1]=1

The Pythagorean equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-ealuoj
Out[1]=1

Bounded systems of equations and inequalities:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-jqvrli
Out[1]=1

High‐degree systems with no solutions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-jjwul
Out[1]=1

Transcendental Diophantine systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-b6qb0s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-j84jjc
Out[2]=2

Polynomial systems of congruences:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cpi4z8
Out[1]=1

Modular Equations  (4)

Linear systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cud08z
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-7cyoa1
Out[2]=2

Univariate polynomial equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-7mc9q
Out[1]=1

Systems of polynomial equations and inequations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-dxz908
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-j4xh3c
Out[2]=2

Quantified polynomial systems:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-kzgfqy
Out[1]=1

Equations over Finite Fields  (3)

Univariate equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cfndqf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-f6gotk
Out[3]=3

Systems of linear equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-hef9nj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-dyppb
Out[2]=2

Systems of polynomial equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fi2b3p
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-fna1o8
Out[2]=2

Systems with Mixed-Variable Domains  (2)

Mixed real and complex variables:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-mc1m3r
Out[1]=1

Mixed real and integer variables:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-bd6p0y
Out[1]=1

Systems with Geometric Region Constraints  (9)

Solve over special regions in 2D:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-bk1i0g
In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-m07wn0
Out[3]=3

Plot it:

In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-okukv
Out[4]=4

Solve over special regions in 3D:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-gnofuq
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-h539x7
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-edimni
Out[3]=3

A quantified system:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-b9wofb
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-chs7bp
Out[2]=2

An implicitly defined region:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cstveu
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-kb9rdr
Out[2]=2

A parametrically defined region:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-juwcyb
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-npl878
Out[2]=2

Derived regions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-mi7sv6
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-bqp6b6
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-la3oq9
Out[3]=3

Eliminate quantifiers over a Cartesian product of regions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-b2oibr
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-fd3wao
Out[2]=2

Regions dependent on parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-9d869w

The answer depends on the parameter value :

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-miz9f3
Out[2]=2

Use to specify that is a vector in :

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cnu18n
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-f27c71
Out[2]=2

In this case, is a vector in :

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-i3d948
In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-hjca7y
Out[4]=4

Options  (26)Common values & functionality for each option

Assumptions  (4)

Specify conditions on parameters using Assumptions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-uoghsw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-ogmx49
Out[2]=2

By default, no solutions that require parameters to satisfy equations are produced:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-0qit7y
Out[1]=1

With an equation on parameters given as an assumption, a solution is returned:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-iauitl
Out[2]=2

Assumptions that contain solve variables are considered to be a part of the system to solve:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-t7553b
Out[1]=1

Equivalent statement without using Assumptions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-slsi1k
Out[2]=2

With parameters assumed to belong to a discrete set, solutions involving arbitrary conditions are returned:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-s81tpo
Out[1]=1

Cubics  (3)

By default, SolveValues uses general formulas for solving cubics in radicals only when symbolic parameters are present:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-evm2xk
Out[1]=1

For polynomials with numeric coefficients, SolveValues does not use the formulas:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-ej107d
Out[2]=2

With Cubics->False, SolveValues never uses the formulas:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-jxrryz
Out[1]=1

With Cubics->True, SolveValues always uses the formulas:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-drg6po
Out[1]=1

GeneratedParameters  (1)

SolveValues may introduce new parameters to represent the solution:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cmun55
Out[1]=1

Use GeneratedParameters to control how the parameters are generated:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-cwwkfe
Out[2]=2

InverseFunctions  (3)

By default, SolveValues uses inverse functions but prints warning messages:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-kghb2b
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-h4qcp8
Out[2]=2

For symbols with the NumericFunction attribute, symbolic inverses are not used:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-hg0smw
Out[3]=3

With InverseFunctions->True, SolveValues does not print inverse function warning messages:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cryml7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-gob22e
Out[2]=2

Symbolic inverses are used for all symbols:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-ffxteo
Out[3]=3

With InverseFunctions->False, SolveValues does not use inverse functions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cnj7jt
Out[1]=1

Solving algebraic equations does not require using inverse functions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-g375m9
Out[2]=2

Here, a method based on Reduce is used, as it does not require using inverse functions:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-dm14cy
Out[3]=3

MaxExtraConditions  (4)

By default, no solutions requiring extra conditions are produced:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-2fqw4
Out[1]=1

Unless the parameters are discrete:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-b6z7v
Out[2]=2

The default setting, MaxExtraConditions->0, gives no solutions requiring conditions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-irkajs
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-i758z1
Out[2]=2

MaxExtraConditions->1 gives solutions requiring up to one equation on parameters:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-bfzk7f
Out[3]=3

MaxExtraConditions->2 gives solutions requiring up to two equations on parameters:

In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-7yi6l
Out[4]=4

Give solutions requiring the minimal number of parameter equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-r58a3v
Out[1]=1

Give all solutions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-vwfwex
Out[2]=2

By default, SolveValues drops inequation conditions on continuous parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-dnfv8h
Out[1]=1

With MaxExtraConditions->All, SolveValues includes all conditions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-i5f1qz
Out[2]=2

MaxRoots  (4)

Find out of roots of a polynomial:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-t6ayi6
Out[1]=1

Find out of roots of a polynomial system:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-etqwxz
Out[1]=1

Find solutions of a transcendental system:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-wv1dky
Out[1]=1

When the system contains symbolic parameters, the option value is ignored:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-3zlejt
Out[1]=1

Method  (1)

By default, SolveValues uses inverse functions to solve non-polynomial complex equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-43b19q
Out[1]=1

With Method->Reduce, SolveValues uses Reduce to find the complete solution set:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-0a608r
Out[2]=2

Modulus  (1)

Solve equations over the integers modulo 9:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-ek6n83
Out[1]=1

Quartics  (3)

By default, SolveValues uses the general formulas for solving quartics in radicals only when symbolic parameters are present:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-ggptus
Out[1]=1

For polynomials with numeric coefficients, SolveValues does not use the formulas:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-x0nf3
Out[2]=2

With Quartics->False, SolveValues never uses the formulas:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-diye9s
Out[1]=1

With Quartics->True, SolveValues always uses the formulas:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cnyrg5
Out[1]=1

VerifySolutions  (1)

SolveValues verifies solutions obtained using non-equivalent transformations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-1czt1
Out[1]=1

With VerifySolutions->False, SolveValues does not verify the solutions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-fh9f80
Out[2]=2

Some of the solutions returned with VerifySolutions->False are not correct:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-drfo0b
Out[3]=3

This uses a fast numeric test in an attempt to select correct solutions:

In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-dn6k7b
Out[4]=4

In this case, numeric verification gives the correct solution set:

In[5]:=5
✖
https://wolfram.com/xid/0bdou600i-fyu6as
Out[5]=5

WorkingPrecision  (1)

By default, SolveValues finds exact solutions of equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-bw43lc
Out[1]=1

Computing the solution using 100-digit numbers is faster:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-fg6bw
Out[2]=2

The result agrees with the exact solution in the first 100 digits:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-gnzq7x
Out[3]=3

Computing the solution using machine numbers is much faster:

In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-fv13j
Out[4]=4

The result is still quite close to the exact solution:

In[5]:=5
✖
https://wolfram.com/xid/0bdou600i-dt8nt1
Out[5]=5

Applications  (7)Sample problems that can be solved with this function

Solve a quadratic equation:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-e5nov5
Out[1]=1

Find intersection points of a circle and a parabola:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-hadad7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-cnnpir
Out[2]=2

Find conditions for a quartic to have all roots equal:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-dm5wo0

A method using Subresultants:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-by6uvl
Out[2]=2

A method using quantifier elimination:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-dp4mfy
Out[3]=3

Plot a space curve given by an implicit description:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-isi6u
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-b5j5yf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-gcg9qy
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-9qhiyb
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bdou600i-esadtv
Out[5]=5

Plot the projection of the space curve on the {x,y} plane:

In[6]:=6
✖
https://wolfram.com/xid/0bdou600i-g489md
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0bdou600i-e00aj1
Out[7]=7

Find a Pythagorean triple:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-eylfw
Out[1]=1

Find a sequence of Pythagorean triples:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-fko26k
Out[2]=2

Find how to pay $2.27 postage with 10-, 23-, and 37-cent stamps:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fw1flg
Out[1]=1

The same task can be accomplished with IntegerPartitions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-hzps68
Out[2]=2

Find 200 roots of a complex analytic function:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-4xm7wy

Show the roots on the complex plot for the function:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-6rl2pu
Out[2]=2

Properties & Relations  (15)Properties of the function, and connections to other functions

Solutions are given as lists and satisfy the equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-044h5
In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cek3yh
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-dhosxz
Out[3]=3

For univariate equations, SolveValues repeats solutions according to their multiplicity:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-n4r5o7
Out[1]=1

Solutions of algebraic equations are often given in terms of Root objects:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-hzxpz4
Out[1]=1

Use N to compute numeric approximations of Root objects:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-oe2s7b
Out[2]=2

Root objects may involve parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-houqyv
Out[1]=1

Use Series to compute series expansions of Root objects:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-mr0
Out[2]=2

The series satisfies the equation up to order 11:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-ud261
Out[3]=3

SolveValues gives values of the solutions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-lggsq3
Out[1]=1

Solve represents solutions in terms of replacement rules:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-b3ttxk
Out[2]=2

Reduce represents solutions in terms of Boolean combinations of equations and inequalities:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-foau3
Out[3]=3

SolveValues uses fast heuristics to solve transcendental equations, but may give incomplete solutions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-hj9p8a
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-sq8nip
Out[2]=2

Reduce uses methods that are often slower, but finds all solutions and gives all necessary conditions:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-e1w6vc
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-bfwk75
Out[4]=4

Use FindInstance to find solution instances:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-buc7ih
Out[1]=1

Like Reduce, FindInstance can be given inequalities and domain specifications:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-bdrytq
Out[2]=2

Use DSolve to solve differential equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fq4m7y
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-ctk70u
Out[2]=2

Use RSolve to solve recurrence equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cmoc60
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-mdx5f
Out[2]=2

SolveAlways gives the values of parameters for which complex equations are always true:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-d84317
Out[1]=1

The same problem can be expressed using ForAll and solved with SolveValues, Solve or Reduce:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-5n5rp
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-i9ebg2
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-s758q1
Out[4]=4

Resolve eliminates quantifiers, possibly without solving the resulting quantifier-free system:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-nq0ez
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-gsmoto
Out[2]=2

Eliminate eliminates variables from systems of complex equations:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-l19n1
Out[1]=1

This solves the same problem using Resolve:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-mvxma
Out[2]=2

Reduce, Solve and SolveValues additionally solve the resulting equations:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-ekfiuo
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-jlsuow
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bdou600i-itshg4
Out[5]=5

is bijective iff the equation has exactly one solution for each :

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-ec0f5e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-h9imdc
Out[2]=2

Use FunctionBijective to test whether a function is bijective:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-fgftkt
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-bgha30
Out[4]=4

Use FunctionAnalytic to test whether a function is analytic:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-fflilk
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-ccc4nd
Out[2]=2

An analytic function can have only finitely many zeros in a closed and bounded region:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-60azb
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-gjv4ax
Out[4]=4

SolveValues finds an explicit function of satisfying the equation :

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-cgm0xt
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-b78e1
Out[2]=2

Use ImplicitD to find the derivative of an implicitly defined function:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-26s55
Out[3]=3

Possible Issues  (9)Common pitfalls and unexpected behavior

SolveValues gives generic solutions; solutions involving equations on parameters are not given:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-chgpbe
Out[1]=1

Reduce gives all solutions, including those that require equations on parameters:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-mlktc
Out[2]=2

With MaxExtraConditions->All, SolveValues also gives non-generic solutions:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-8i223l
Out[3]=3

SolveValues results do not depend on whether some of the input equations contain only parameters. The following two systems are equivalent and have no generic solutions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-l7ztbs
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-v9s0d
Out[2]=2

Use MaxExtraConditions to specify the number of parameter conditions allowed:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-cwhzx7
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-gbhbxg
Out[4]=4

Use the Exists quantifier to find solutions that are valid for some value of parameter a:

In[5]:=5
✖
https://wolfram.com/xid/0bdou600i-iya3dm
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bdou600i-buse5o
Out[6]=6

SolveValues does not eliminate solutions that are neither generically correct nor generically incorrect:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-j4lv55
Out[1]=1

The solutions are correct for and incorrect for :

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-blev95
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-jy34mb
Out[3]=3

For transcendental equations, SolveValues may not give all solutions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-ea501u
Out[1]=1

Use Reduce to get all solutions:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-e9it8x
Out[2]=2

SolveValues with Method->"Reduce" uses Reduce to find solutions, but returns solution values:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-qudqup
Out[3]=3

Using inverse functions allows SolveValues to find some solutions fast:

In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-cnjb7c
Out[4]=4

Finding the complete solution may take much longer, and the solution may be large:

In[5]:=5
✖
https://wolfram.com/xid/0bdou600i-dtv6ol
Out[5]=5

This finds the values of n for which x==2 is a solution:

In[6]:=6
✖
https://wolfram.com/xid/0bdou600i-lafmqw
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0bdou600i-c6fxzo
Out[7]=7

Interpretation of assumptions depends on their syntactic properties. Here the solution is generic in the parameter space restricted by the assumptions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-gjc9eb
Out[1]=1

This mathematically equivalent assumption contains the solve variable, and hence is treated as a part of the system to solve:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-pueeg0
Out[2]=2

There are no generic solutions, because the input is interpreted as:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-fo41q0
Out[3]=3

The solution is non-generic, since it requires the parameters to satisfy an equation:

In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-dkivdq
Out[4]=4

When parameters are restricted to a discrete set, the notion of genericity is not well defined, and all solutions are returned:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-v48g2c
Out[1]=1

Removable singularities of input equations are generally not considered valid solutions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-ssq1n2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-zows00
Out[2]=2

However, solutions may include removable singularities that are cancelled by automatic simplification:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-2vcfa3
Out[3]=3

The removable singularity at is cancelled by evaluation:

In[4]:=4
✖
https://wolfram.com/xid/0bdou600i-1xtv8x
Out[4]=4

Here the removable singularity at is cancelled by Together, which is used to preprocess the equation:

In[5]:=5
✖
https://wolfram.com/xid/0bdou600i-you5oc
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bdou600i-dmxlbo
Out[6]=6

The value of MaxRoots is used only for systems with numeric coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-f9fir
Out[1]=1

When symbolic parameters are present, the option value is ignored:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-jt20ld
Out[2]=2

Expressions given as variables are treated as atomic objects and not as functions of their subexpressions:

In[1]:=1
✖
https://wolfram.com/xid/0bdou600i-ii8ifm
Out[1]=1

Effectively, variables are replaced with new symbols before the equations are solved:

In[2]:=2
✖
https://wolfram.com/xid/0bdou600i-z4gwo6
Out[2]=2

The result comes from:

In[3]:=3
✖
https://wolfram.com/xid/0bdou600i-f7wm6w
Out[3]=3
Wolfram Research (2021), SolveValues, Wolfram Language function, https://reference.wolfram.com/language/ref/SolveValues.html (updated 2024).
✖
Wolfram Research (2021), SolveValues, Wolfram Language function, https://reference.wolfram.com/language/ref/SolveValues.html (updated 2024).

Text

Wolfram Research (2021), SolveValues, Wolfram Language function, https://reference.wolfram.com/language/ref/SolveValues.html (updated 2024).

✖
Wolfram Research (2021), SolveValues, Wolfram Language function, https://reference.wolfram.com/language/ref/SolveValues.html (updated 2024).

CMS

Wolfram Language. 2021. "SolveValues." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/SolveValues.html.

✖
Wolfram Language. 2021. "SolveValues." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/SolveValues.html.

APA

Wolfram Language. (2021). SolveValues. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SolveValues.html

✖
Wolfram Language. (2021). SolveValues. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SolveValues.html

BibTeX

@misc{reference.wolfram_2025_solvevalues, author="Wolfram Research", title="{SolveValues}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/SolveValues.html}", note=[Accessed: 27-March-2025 ]}

✖
@misc{reference.wolfram_2025_solvevalues, author="Wolfram Research", title="{SolveValues}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/SolveValues.html}", note=[Accessed: 27-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_solvevalues, organization={Wolfram Research}, title={SolveValues}, year={2024}, url={https://reference.wolfram.com/language/ref/SolveValues.html}, note=[Accessed: 27-March-2025 ]}

✖
@online{reference.wolfram_2025_solvevalues, organization={Wolfram Research}, title={SolveValues}, year={2024}, url={https://reference.wolfram.com/language/ref/SolveValues.html}, note=[Accessed: 27-March-2025 ]}