Built-in Mathematica Symbol

Solve

Solve[eqns, vars]
attempts to solve an equation or set of equations for the variables vars.

Solve[eqns, vars, elims]
attempts to solve the equations for vars, eliminating the variables elims.

MORE INFORMATION

Equations are given in the form lhsrhs.

Simultaneous equations can be combined either in a list or with &&.

A single variable or a list of variables can be specified.

Solve[eqns] tries to solve for all variables in eqns.

Solve gives solutions in terms of rules of the form x->sol.

When there are several variables, the solution is given in terms of lists of rules: {x->s_x, y->s_y, ...}.

When there are several solutions, Solve gives a list of them.

When a particular root has multiplicity greater than one, Solve gives several copies of the corresponding solution. »

Solve deals primarily with linear and polynomial equations.

The option InverseFunctions specifies whether Solve should use inverse functions to try and find solutions to more general equations. The default is InverseFunctions->Automatic. In this case, Solve can use inverse functions, but prints a warning message. See notes on InverseFunctions. »

Solve gives generic solutions only. It discards solutions that are valid only when the parameters satisfy special conditions. Reduce gives the complete set of solutions. »

Solve will not always be able to get explicit solutions to equations. It will give the explicit solutions it can, then give a symbolic representation of the remaining solutions in terms of Root objects. If there are sufficiently few symbolic parameters, you can then use N to get numerical approximations to the solutions. »

Solve gives {} if there are no possible solutions to the equations. »

Solve gives {{}} if all variables can have all possible values. »

Solve[eqns, ..., Mode->Modular] solves equations with equality required only modulo an integer. You can specify a particular modulus to use by including the equation Modulusp. If you do not include such an equation, Solve will attempt to solve for the possible moduli. »

Solve uses special efficient techniques for handling sparse systems of linear equations with approximate numerical coefficients.

EXAMPLES

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Basic Examples (4)

Solve a quadratic equation:

Solve simultaneous equations in x and y:

Solutions are given as lists of replacements:

Replace x by solutions:

Replace combinations of x and y by solutions, and simplify the results:

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

Pick out the third solution:

Solve a quadratic equation:

In[1]:=

Out[1]=

Solve simultaneous equations in x and y:

In[1]:=

Out[1]=

Solutions are given as lists of replacements:

In[1]:=

Out[1]=

Replace x by solutions:

In[2]:=

Out[2]=

Replace combinations of x and y by solutions, and simplify the results:

In[3]:=

Out[3]=

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

In[4]:=

Out[4]=

Pick out the third solution:

Scope (18)

Polynomial equations solvable in radicals:

General polynomial equations:

Polynomial equations with multiple roots:

Polynomial equations with symbolic coefficients:

Algebraic equations:

Transcendental equations:

Symbolic functions:

Systems of linear equations:

Linear equations with symbolic coefficients:

Underdetermined systems of linear equations:

Linear equations with no solutions:

Systems of polynomial equations:

Polynomial equations with symbolic coefficients:

Algebraic equations:

Transcendental equations:

A linear system:

Univariate polynomial equation:

A system of polynomial equations and inequations:

Generalizations & Extensions (1)

All variables are solved for:

Options (9)

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

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

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

Solving algebraic equations does not require using inverse functions:

Find a modulus for which a system of equations has a solution:

By default, Solve heuristically selects a variable order which makes the problem easier:

This forces Solve to find solutions for y first, and solve for x in terms of these solutions:

Solve verifies solutions obtained using non-equivalent transformations:

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

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

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

In this case numeric verification gives the correct solution set:

By default, Solve does not verify solutions of equations with inexact coefficients:

In some of the solutions returned, polys are quite far from zero:

With VerifySolutions->True, Solve increases the precision of computations when needed:

The values of polys at the solutions are now much closer to zero:

By default, Solve finds exact solutions of equations:

Computing the solution using 100-digit numbers is faster:

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

Computing the solution using machine numbers is much faster:

The result is still quite close to the exact solution:

Applications (2)

Solve a quadratic equation:

Find intersection points of a circle and a parabola:

Properties & Relations (11)

Solutions satisfy the equations:

Solutions are given as replacement rules and can be directly used for substitution:

Solve uses {} to represent the empty or no solution:

Solve uses {{}} to represent the universal solution or that all points satisfy the equations:

Solve will produce solutions according to their multiplicity:

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

Use N to compute numeric approximations of Root objects:

Root objects may involve parameters:

Use Series to compute series expansions of Root objects:

The series satisfies the equation up to order 11:

Solve represents solutions in terms of replacement rules:

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

Use Reduce to solve equations and inequalities over specified domains:

Use FindInstance to find solution instances:

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

Use DSolve to solve differential equations:

Use RSolve to solve recurrence equations:

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

This solves the same problem using Reduce:

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

Eliminate eliminates variables from systems of complex equations:

This solves the same problem using Resolve:

Reduce additionally solves the resulting equations:

Possible Issues (5)

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

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

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

The solutions are correct for

and incorrect for

:

Input equations involving only parameters are treated as assumptions:

Without the assumption on parameters Solve gives no solutions:

Reduce gives the solution together with the condition on parameters:

For transcendental equations Solve may not give all solutions:

Using inverse functions allows Solve to find some solutions fast:

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

This finds the values of n for which x

2 is a solution:

A polynomial modular equation with symbolic coefficients generates an implicit solution:

Get explicit solutions: