You can solve an equation using Solve. Remember to use "" in an equation, not just "=":

The result is a Rule inside a doubly nested list. The outer list holds all of the solutions and each inner list holds a single solution. Here there are three solutions:

To solve a system of equations, use a list in the first argument:

Here there are two solutions to a simultaneous system of equations; each solution set is wrapped in its own list:

Here the solution expresses one variable in terms of another:

To use one of these solutions (here the first one is shown), use [[...]] (the short form of Part) to extract it from the list of solutions and use /. (the short form of ReplaceAll) to apply the rule:

For example, here is a plot of x^2-y^2 for different values of y, assuming that the first solution holds:

In a system of equations with multiple variables, you can solve for some or all of the variables by using a list in the second argument:

If the system is underspecified, Mathematica will give an answer in terms of the remaining variables:

Solve finds what are known as "generic" solutions to equations. These are solutions that do not depend on the variables not specified in the second argument. For example:

No matter what y is, putting in 0 for x solves the equation. But there is another solution that does depend on y: namely, setting y to 0. Adding y in the second argument makes this solution show up:

There are other cases in which Solve does not find every solution. For example:

You can also solve equations by using Reduce:

The output of Reduce is different from the output of Solve: Reduce outputs a logical expression that is equivalent to the original equation, so it never omits a solution: