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FindInstance

 FindInstance[expr, vars]finds an instance of vars that makes the statement expr be True. FindInstance[expr, vars, dom]finds an instance over the domain dom. Common choices of dom are Complexes, Reals, Integers and Booleans. FindInstance[expr, vars, dom, n]finds n instances.
• FindInstance[expr, {x1, x2, ...}] gives results in the same form as Solve: {{x1->val1, x2->val2, ...}} if an instance exists, and {} if it does not.
• expr can contain equations, inequalities, domain specifications and quantifiers, in the same form as in Reduce.
• With exact symbolic input, FindInstance gives exact results.
• Even if two inputs define the same mathematical set, FindInstance may still pick different instances to return.
• The instances returned by FindInstance typically correspond to special or interesting points in the set.
• FindInstance[expr, vars] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
• FindInstance[expr, vars, Reals] assumes that not only vars but also all function values in expr are real. FindInstance[expr&&varsReals, vars] assumes only that the vars are real.
• FindInstance may be able to find instances even if Reduce cannot give a complete reduction.
• Every time you run FindInstance with a given input, it will return the same output.
• Different settings for the option RandomSeed->s may yield different collections of instances.
• FindInstance[expr, vars, dom, n] will return a shorter list if the total number of instances is less than n.
Find a solution instance of a system of equations:
Find a real solution instance of a system of equations and inequalities:
Find an integer solution instance:
Find Boolean values of variables that satisfy a formula:
Find several instances:
Find a solution instance of a system of equations:
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Find a real solution instance of a system of equations and inequalities:
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Find an integer solution instance:
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Find Boolean values of variables that satisfy a formula:
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Find several instances:
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 Scope   (41)
A linear system:
A univariate polynomial equation:
A multivariate polynomial equation:
Systems of polynomial equations and inequations:
This gives three solution instances:
If there are no solutions FindInstance returns an empty list:
If there are fewer solutions than the requested number, FindInstance returns all solutions:
Quantified polynomial system:
An algebraic system:
Transcendental equations:
In this case there is no solution:
A solution in terms of transcendental Root objects:
A system of transcendental equations:
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:
Get four solution instances:
If there are no solutions FindInstance returns an empty list:
If there are fewer solutions than the requested number, FindInstance returns all solutions:
A quantified polynomial system:
An algebraic system:
Piecewise equations:
Piecewise inequalities:
Transcendental equations:
A solution in terms of transcendental Root objects:
Transcendental inequalities:
Transcendental systems:
A linear system of equations:
A linear system of equations and inequalities:
Find more than one solution:
A univariate polynomial equation:
A univariate polynomial inequality:
A Thue equation:
If there are fewer solutions than the requested number, FindInstance returns all solutions:
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:
A linear system:
A univariate polynomial equation:
A multivariate polynomial equation:
Find seven instances:
A system of polynomial equations and inequations:
A quantified polynomial system:
Mixed real and complex variables:
Find a real value of and a complex value of for which is real and less than :
An inequality involving Abs[z]:
 Options   (3)
Find a solution over the integers modulo 9:
Find three solutions:
Finding instances often involves random choice from large solution sets:
By default, FindInstance chooses the same solutions each time:
With a different RandomSeed, FindInstance may give different solutions:
Finding an exact solution to this problem is hard due to high degrees of algebraic numbers:
With a finite WorkingPrecision, FindInstance is able to find an approximate solution:
 Applications   (6)
Find a point in the intersection of two regions:
Find a counterexample to a geometric conjecture:
Prove the conjecture using stronger assumptions:
Prove that a statement is a tautology:
This can be proven with TautologyQ as well:
Show that a statement is not a tautology; get a counterexample:
This can be done with SatisfiabilityInstances as well:
Find a Pythagorean triple:
Find Pythagorean triples when they exist:
Two instances are now found when :
Show that there are no 2×2 magic squares with all numbers unequal:
Solution instances satisfy the input system:
Use RootReduce to prove that algebraic numbers satisfy equations:
When there are no solutions, FindInstance returns an empty list:
If there are fewer solutions than the requested number, FindInstance returns all solutions:
To get a complete description of the solution set use Reduce:
To get a generic solution of a system of complex equations use Solve:
Solving a sum of squares representation problem:
Use SquaresR to find the number of solutions to sum of squares problems:
Solving a sum of powers representation problem:
Use PowersRepresentations to enumerate all solutions:
Find instances satisfying a Boolean statement:
Use SatisfiabilityInstances to obtain solutions represented as Boolean vectors
Integer solutions for a Thue equation:
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