"Defining Variables" discussed assignments such as x=y, which set x equal to y. Here we discuss equations, which test equality. The equation x==y tests whether x is equal to y.

It is very important that you do not confuse x=y with x==y. While x=y is an imperative statement that actually causes an assignment to be done, x==y merely tests whether x and y are equal, and causes no explicit action. If you have used the C programming language, you will recognize that the notation for assignment and testing in the Wolfram Language is the same as in C.

The tests used so far involve only numbers, and always give a definite answer, either True or False. You can also do tests on symbolic expressions.

Even when you do tests on symbolic expressions, there are some cases where you can get definite results. An important one is when you test the equality of two expressions that are identical. Whatever the numerical values of the variables in these expressions may be, the Wolfram Language knows that the expressions must always be equal.

Expressions like x==4 represent equations in the Wolfram Language. There are many functions in the Wolfram Language for manipulating and solving equations.

An expression like x^2+2x-7==0 represents an equation in the Wolfram Language. You will often need to solve equations like this, to find out for what values of x they are true.

Solve always tries to give you explicit formulas for the solutions to equations. However, it is a basic mathematical result that, for sufficiently complicated equations, explicit algebraic formulas in terms of radicals cannot be given. If you have an algebraic equation in one variable, and the highest power of the variable is at most four, then the Wolfram Language can always give you formulas for the solutions. However, if the highest power is five or more, it may be mathematically impossible to give explicit algebraic formulas for all the solutions.

In addition to being able to solve purely algebraic equations, the Wolfram Language can also solve some equations involving other functions.

It is important to realize that an equation such as actually has an infinite number of possible solutions. However, Solve by default returns just one solution, but prints a message telling you that other solutions may exist. You can use Reduce to get more information.

Solve can also handle equations involving symbolic functions. In such cases, it again prints a warning, then gives results in terms of formal inverse functions.

You can also use the Wolfram Language to solve sets of simultaneous equations. You simply give the list of equations and specify the list of variables to solve for.

The Wolfram Language can solve any set of simultaneous linear or polynomial equations.

When you are working with sets of equations in several variables, it is often convenient to reorganize the equations by eliminating some variables between them.

If you have several equations, there is no guarantee that there exists any consistent solution for a particular variable.

The general question of whether a set of equations has any consistent solution is quite a subtle one. For example, for most values of a, the equations {x==1,x==a} are inconsistent, so there is no possible solution for x. However, if a is equal to 1, then the equations do have a solution. Solve is set up to give you generic solutions to equations. It discards any solutions that exist only when special constraints between parameters are satisfied.

If you use Reduce instead of Solve, the Wolfram Language will, however, keep all the possible solutions to a set of equations, including those that require special conditions on parameters.

Reduce also has powerful capabilities for handling equations specifically over real numbers or integers. "Equations and Inequalities over Domains" discusses this in more detail.

The Wolfram System treats equations as logical statements. If you type in an equation like x^2+3x==2, the Wolfram System interprets this as a logical statement that asserts that x^2+3x is equal to 2. If you have assigned an explicit value to x, say x=4, then the Wolfram System can explicitly determine that the logical statement x^2+3x==2 is False.

If you have not assigned any explicit value to x, however, the Wolfram System cannot work out whether x^2+3x==2 is True or False. As a result, it leaves the equation in the symbolic form x^2+3x==2.

You can manipulate symbolic equations in the Wolfram System in many ways. One common goal is to rearrange the equations so as to "solve" for a particular set of variables.

The quadratic equation x^2+3x==2 can be thought of as an implicit statement about the value of x. As shown in the previous example, you can use the function Reduce to get a more explicit statement about the value of x. The expression produced by Reduce has the form x==r₁||x==r₂. This expression is again a logical statement, which asserts that either x is equal to r₁, or x is equal to r₂. The values of x that are consistent with this statement are exactly the same as the ones that are consistent with the original quadratic equation. For many purposes, however, the form that Reduce gives is much more useful than the original equation.

You can combine and manipulate equations just like other logical statements. You can use logical connectives such as || and && to specify alternative or simultaneous conditions. You can use functions like LogicalExpand, as well as FullSimplify, to simplify collections of equations.

For many purposes, you will find it convenient to manipulate equations simply as logical statements. Sometimes, however, you will actually want to use explicit solutions to equations in other calculations. In such cases, it is convenient to convert equations that are stated in the form lhs==rhs into transformation rules of the form lhsrhs. Once you have the solutions to an equation in the form of explicit transformation rules, you can substitute the solutions into expressions by using the /. operator.

The main equations that Solve and related Wolfram Language functions deal with are polynomial equations.

For cubic and quartic equations, the results are often complicated, but for all equations with degrees up to four the Wolfram Language is always able to give explicit formulas for the solutions.

An important feature of these formulas is that they involve only radicals: arithmetic combinations of square roots, cube roots and higher roots.

It is a fundamental mathematical fact, however, that for equations of degree five or higher, it is no longer possible in general to give explicit formulas for solutions in terms of radicals.

There are some specific equations for which this is still possible, but in the vast majority of cases it is not.

Root objects provide an exact, though implicit, representation for the roots of a polynomial. You can work with them much as you would work with Sqrt[2] or any other expression that represents an exact numerical quantity.

If the only symbolic parameter that exists in an equation is the variable that you are solving for, then all the solutions to the equation will just be numbers. But if there are other symbolic parameters in the equation, then the solutions will typically be functions of these parameters.

If you give Solve any ^th‐degree polynomial equation, then it will always return exactly solutions, although some of these may be represented by Root objects. If there are degenerate solutions, then the number of times that each particular solution appears will be equal to its multiplicity.

The Wolfram Language also knows how to solve equations that are not explicitly in the form of polynomials.

So long as it can reduce an equation to some kind of polynomial form, the Wolfram Language will always be able to represent its solution in terms of Root objects. However, with more general equations, involving, say, transcendental functions, there is no systematic way to use Root objects, or even necessarily to find numerical approximations.

Polynomial equations in one variable only ever have a finite number of solutions. But transcendental equations often have an infinite number. Typically the reason for this is that transcendental functions in effect have infinitely many possible inverses. With the default option setting InverseFunctionsTrue, Solve will nevertheless assume that there is a definite inverse for any such function. Solve may then be able to return particular solutions in terms of this inverse function.

If you ask Solve to solve an equation involving an arbitrary function like f, it will by default try to construct a formal solution in terms of inverse functions.

While Solve can only give specific solutions to an equation, Reduce can give a representation of a whole solution set. For transcendental equations, it often ends up introducing new parameters, say with values ranging over all possible integers.

As discussed at more length in "Equations and Inequalities over Domains", Reduce allows you to restrict the domains of variables. Sometimes this will let you generate definite solutions to transcendental equations—or show that they do not exist.

CountRoots accepts polynomials with Gaussian rational coefficients. The root count includes multiplicities.

A set , where is or , is an isolating set for a root of a polynomial if is the only root of in . Isolating roots of a polynomial means finding disjoint isolating sets for all the roots of the polynomial.

RootIntervals accepts polynomials with rational number coefficients.

For a real root , the returned isolating interval is a pair of rational numbers , such that either or . For a nonreal root , the isolating rectangle returned is a pair of Gaussian rational numbers such that and either or .

Root objects are the way that the Wolfram Language represents algebraic numbers. Algebraic numbers have the property that when you perform algebraic operations on them, you always get a single algebraic number as the result.

If Solve and ToRadicals do not succeed in expressing the solution to a particular polynomial equation in terms of radicals, then it is a good guess that this fundamentally cannot be done. However, you should realize that there are some special cases in which a reduction to radicals is in principle possible, but the Wolfram System cannot find it. The simplest example is the equation , but here the solution in terms of radicals is very complicated. The equation is another example, where now is a solution.

Beyond degree four, most polynomials do not have roots that can be expressed at all in terms of radicals. However, for degree five it turns out that the roots can always be expressed in terms of elliptic or hypergeometric functions. The results, however, are typically much too complicated to be useful in practice.

You can give Solve a list of simultaneous equations to solve. Solve can find explicit solutions for a large class of simultaneous polynomial equations.

The variables that you use in Solve do not need to be single symbols. Often when you set up large collections of simultaneous equations, you will want to use expressions like a[i] as variables.

In some kinds of computations, it is convenient to work with arrays of coefficients instead of explicit equations. You can construct such arrays from equations by using CoefficientArrays.

If you have an equation like 2x==0, it is perfectly clear that the only possible solution is x0. However, if you have an equation like ax==0, things are not so clear. If a is not equal to zero, then x0 is again the only solution. However, if a is in fact equal to zero, then any value of x is a solution. You can see this by using Reduce.

A basic difference between Reduce and Solve is that Reduce gives all the possible solutions to a set of equations, while Solve gives only the generic ones. Solutions are considered "generic" if they involve conditions only on the variables that you explicitly solve for, and not on other parameters in the equations. Reduce and Solve also differ in that Reduce always returns combinations of equations, while Solve gives results in the form of transformation rules.

When you have several simultaneous equations, Reduce can show you under what conditions the equations have solutions. Solve shows you whether there are any generic solutions.

When you work with systems of linear equations, you can use Solve to get generic solutions and Reduce to find out for what values of parameters solutions exist.

For nonlinear equations, the conditions for the existence of solutions can be much more complicated.

When you write down a set of simultaneous equations in the Wolfram Language, you are specifying a collection of constraints between variables. When you use Solve, you are finding values for some of the variables in terms of others, subject to the constraints represented by the equations.

In some cases, you may want to construct explicitly equations in which variables have been eliminated. You can do this using Eliminate.

As a more sophisticated example of Eliminate, consider the problem of writing in terms of the "symmetric polynomials" and .

In dealing with sets of equations, it is common to consider some of the objects that appear as true "variables", and others as "parameters". In some cases, you may need to know for what values of parameters a particular relation between the variables is always satisfied.

You should remember that the logical operations ==, && and || are all double characters in the Wolfram Language. If you have used a programming language such as C, you will be familiar with this notation.

When you give a list of equations to Solve, it assumes that you want all the equations to be satisfied simultaneously. It is also possible to give Solve more complicated logical combinations of equations.

When you use Solve, the final results you get are in the form of transformation rules. If you use Reduce or Eliminate, on the other hand, then your results are logical statements, which you can manipulate further.

The logical statements produced by Reduce can be thought of as representations of the solution set for your equations. The logical connectives &&, || and so on then correspond to operations on these sets.

You may often find it convenient to use special notations for logical connectives, as discussed in "Operators".

Just as the equation x^2+3x==2 asserts that x^2+3x is equal to 2, so also the inequality x^2+3x>2 asserts that x^2+3x is greater than 2. In the Wolfram Language, Reduce works not only on equations, but also on inequalities.

When applied to an equation, Reduce[eqn,x] tries to get a result consisting of simple equations for x of the form x==r₁, …. When applied to an inequality, Reduce[ineq,x] does the exactly analogous thing, and tries to get a result consisting of simple inequalities for x of the form l₁<x<r₁, ….

You can think of the result generated by Reduce[ineq,x] as representing a series of intervals, described by inequalities. Since the graph of a polynomial of degree can go up and down as many as times, a polynomial inequality of degree can give rise to as many as distinct intervals.

Transcendental functions like have graphs that go up and down infinitely many times, so that infinitely many intervals can be generated.

If you have inequalities that involve <= as well as <, there may be isolated points where the inequalities can be satisfied. Reduce represents such points by giving equations.

For inequalities involving several variables, Reduce in effect yields nested collections of interval specifications, in which later variables have bounds that depend on earlier variables.

In geometrical terms, any linear inequality divides space into two halves. Lists of linear inequalities thus define polyhedra, sometimes bounded, sometimes not. Reduce represents such polyhedra in terms of nested inequalities. The corners of the polyhedra always appear among the endpoints of these inequalities.

Lists of inequalities in general represent regions of overlap between geometrical objects. Often the description of these can be quite complicated.