Solving Equations
An expression like
represents an
equation in
Mathematica. You will often need to
solve equations like this, to find out for what values of
they are true.
This gives the two solutions to the quadratic equation
. The solutions are given as replacements for
.
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Here are the numerical values of the solutions.
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You can get a list of the actual solutions for
by applying the rules generated by
Solve to
using the replacement operator.
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You can equally well apply the rules to any other expression involving
.
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| Solve[lhs==rhs,x] | solve an equation, giving a list of rules for x |
| x/.solution | use the list of rules to get values for x |
| expr/.solution | use the list of rules to get values for an expression |
Finding and using solutions to equations.
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
Mathematica 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.
Mathematica can always solve algebraic equations in one variable when the highest power is less than five.
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It can solve some equations that involve higher powers.
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There are some equations, however, for which it is mathematically impossible to find explicit formulas for the solutions.
Mathematica uses
Root objects to represent the solutions in this case.
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Even though you cannot get explicit formulas, you can still evaluate the solutions numerically.
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In addition to being able to solve purely algebraic equations,
Mathematica can also solve some equations involving other functions.
After printing a warning,
Mathematica returns one solution to this equation.
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It is important to realize that an equation such as
actually has an infinite number of possible solutions, in this case differing by multiples of
. 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.
There is no explicit "closed form" solution for a transcendental equation like this.
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You can find an approximate numerical solution using
FindRoot, and giving a starting value for
.
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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.
Mathematica returns a result in terms of the formal inverse function of
.
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| Solve[{lhs1==rhs1,lhs2==rhs2,...},{x,y,...}] |
| solve a set of simultaneous equations for x, y, ... |
Solving sets of simultaneous equations.
You can also use
Mathematica to solve sets of simultaneous equations. You simply give the list of equations, and specify the list of variables to solve for.
Here is a list of two simultaneous equations, to be solved for the variables
and
.
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Here are some more complicated simultaneous equations. The two solutions are given as two lists of replacements for
and
.
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This uses the solutions to evaluate the expression
.
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Mathematica 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.
This eliminates
between the two equations, giving a single equation for
.
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If you have several equations, there is no guarantee that there exists
any consistent solution for a particular variable.
There is no consistent solution to these equations, so
Mathematica returns
, indicating that the set of solutions is empty.
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There is also no consistent solution to these equations for almost all values of
.
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The general question of whether a set of equations has any consistent solution is quite a subtle one. For example, for most values of
, the equations
are inconsistent, so there is no possible solution for
. However, if
is equal to
, 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,
Mathematica will however keep
all the possible solutions to a set of equations, including those that require special conditions on parameters.
This shows that the equations have a solution only when
. The notation
represents the requirement that
both 
and 
should be
True.
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This gives the complete set of possible solutions to the equation. The answer is stated in terms of a combination of simpler equations.
indicates equations that must simultaneously be true;
indicates alternatives.
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This gives a more complicated combination of equations.
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This gives a symbolic representation of all solutions.
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| Solve[lhs==rhs,x] | solve an equation for x |
| Solve[{lhs1==rhs1,lhs2==rhs2,...},{x,y,...}] |
| solve a set of simultaneous equations for x, y, ... |
| Eliminate[{lhs1==rhs1,lhs2==rhs2,...},{x,...}] |
| eliminate x, ... in a set of simultaneous equations |
| Reduce[{lhs1==rhs1,lhs2==rhs2,...},{x,y,...}] |
| give a set of simplified equations, including all possible solutions |
Functions for solving and manipulating equations.
Reduce also has powerful capabilities for handling equations specifically over real numbers or integers.
"Equations and Inequalities over Domains" discusses this in more detail.
This reduces the equation assuming
and
are complex.
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This includes the conditions for
and
to be real.
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This gives only the integer solutions.
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