Find an explicit description of the solution set of a system of equations:

Use ToRules and ReplaceRepeated (//.) to list the solutions:

Find an explicit description of the solution set of a system of inequalities:

Find solutions over specified domains:

The solution set may depend on symbolic parameters:

Representing solutions may require introduction of new parameters:

List the first 10 solutions:

A linear system:

A univariate polynomial equation:

A multivariate polynomial equation:

Systems of polynomial equations and inequations can always be reduced:

A quantified polynomial system:

An algebraic system:

Transcendental equations solvable in terms of inverse functions:

In this case there is no solution:

Equations involving elliptic functions:

Equations solvable using special function zeros:

Solving this system does not require the Riemann hypothesis:

Elementary function equation in a bounded region:

Holomorphic function equation in a bounded region:

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

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

Doubly periodic transcendental equation:

A system of transcendental equations solvable using inverse functions:

A square system of analytic equations over a bounded box:

A linear system:

A univariate polynomial equation:

A univariate polynomial inequality:

A multivariate polynomial equation:

A multivariate polynomial inequality:

Systems of polynomial equations and inequalities can always be reduced:

A quantified polynomial system:

An algebraic system:

Piecewise equations:

Piecewise inequalities:

Transcendental equations, solvable using inverse functions:

Transcendental inequalities, solvable using inverse functions:

Inequalities involving elliptic functions:

Transcendental equation, solvable using special function zeros:

Transcendental inequality, solvable using special function zeros:

Exp-log equations:

High-degree sparse polynomial equation:

Algebraic equation involving high-degree radicals:

Equation involving irrational real powers:

Exp-log inequality:

Elementary function equation in a bounded interval:

Holomorphic function equation in a bounded interval:

Meromorphic function inequality in a bounded interval:

Periodic elementary function equation over the reals:

Transcendental systems solvable using inverse functions:

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

Quantified system:

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

Quantified system:

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

Quantified system:

Square systems of analytic equations over bounded regions:

Linear system of equations:

A linear system of equations and inequalities:

A univariate polynomial equation:

A univariate polynomial inequality:

Binary quadratic equations:

A Thue equation:

A sum of squares equation:

The Pythagorean equation:

A bounded system of equations and inequalities:

A high-degree system with no solution:

Transcendental Diophantine systems:

A polynomial system of congruences:

Diophantine equations with irrational coefficients:

A linear system:

A univariate polynomial equation:

A multivariate polynomial equation:

A system of polynomial equations and inequations:

Reduce a quantified polynomial system:

Univariate equations:

Systems of linear equations:

Systems of polynomial equations:

Systems involving quantifiers:

Mixed real and complex variables:

Find real values of and complex values of for which is real and less than :

Reduce an inequality involving Abs[x]:

Plot the solution set:

Mixed integer and real variables:

Constrain variables to basic geometric regions in 2D:

Plot the solution:

Constrain variables to basic geometric regions in 3D:

Plot the solution:

Project a 3D region onto the - plane:

Plot the projection:

An implicitly defined region:

A parametrically defined region:

Derived regions:

The solution of restricted to the intersection:

Eliminate quantifiers over a Cartesian product of regions:

Regions dependent on parameters:

A condition for :

Use to specify that is a vector in :

In this case is a vector in :

Since y appears after x in the variable list, Reduce may use x to express the solution for y:

With Backsubstitution->True, Reduce gives explicit numeric values for y:

By default, Reduce does not use general formulas for solving cubics in radicals:

With Cubics->True, Reduce solves all cubics in terms of radicals:

Reduce may introduce new parameters to represent the solution:

Use GeneratedParameters to control how the parameters are generated:

Solve equations over the integers modulo 9:

By default, Reduce does not use general formulas for solving quartics in radicals:

With Quartics->True, Reduce solves all quartics in terms of radicals:

Finding the solution with exact computations takes a long time:

With WorkingPrecision->100, Reduce finds a solution fast, but it may be incorrect:

Prove geometric inequalities for , , and sides of a triangle:

Prove an inequality for triangles:

Prove an inequality for acute triangles:

Find conditions for a quartic to have all roots equal:

Using quantifier elimination:

Using Subresultants:

Plot a space curve given by an implicit description:

Plot the projection of the space curve on the - plane:

Find a Pythagorean triple:

Find a sequence of Pythagorean triples:

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

The same task can be accomplished with IntegerPartitions:

Show that there are only five regular polyhedrons:

Each face for a regular -gon contributes edges, but they are shared, so they are counted twice:

Each face for a regular -gon contributes vertices, but they are shared, so they are counted times:

Using Euler's formula , find the number of faces:

For this last formula to be well defined, the denominator needs to be positive and an integer:

Hence the following five cases:

Compare this to the actual counts in PolyhedronData:

The region ℛ is a subset of  if is true. Show that Disk[{0,0},{2,1}] is a subset of Rectangle[{-2,-1},{2,1}]:

Plot it:

Show that Cylinder[]⊆Ball[{0,0,0},2]:

Plot it:

For a finite point set , the Voronoi cell for a point can be defined by , which corresponds to all points closer to than any other point for . Find a simple formula for a Voronoi cell, using Reduce:

The Voronoi cell associated with pts〚1〛 is given by:

The resulting cell is given by an intersection of half-spaces:

Find simple formulas for all Voronoi cells:

Plot them: