Q:

In the context of mathematics, What is the difference between equation and formula?

Equation and formula, these 2 terms are used alternatively in "Introduction to Linear Algebra 4th edition" without giving a clear definition, so, what is the difference?
Can any one provide some concrete examples like following, to distinguish these 2 terms, especially a case is a formula rather than an equation.

Even a supercomputer doesn't want the inverse matrix: too slow.
  Inverses give the simplest formula $x = A^{-1}$ b but not the top speed.
  And everyone must know that determinants are even slower-there is no
  way a linear algebra course should begin with formulas for the
  determinant of an n by n matrix. Those formulas have a place, but not
  first place.
The equation Ax = b uses the language of linear combinations right
  away.

This question is different to this post, which gives a lot of non-math examples without giving a clear comparison between the 2 terms in the context of mathematics.

A:

An equation is a statement of equality, however latter is defined. The usual symbol in mathematics abbreviating is equal to is $=.$ Thus, any formula containing that symbol is called an equation.
That brings us to formula. A formula is an expression, but it's usually used for well-formed expressions. From the name, a formula is a recipe -- sort of -- a method, an algorithm, expressing a sequence of operations and relationships. Thus all equations may be regarded as formulae, but clearly there are many other expressions that are not equations. We may make a distinction between a formula and its expression, but I have blurred that distinction here, just as we blur the distinction between a numeral and a number, or between a function and its values, for example.

A:

A formula is any expression built from mathematical symbols built in accordance to their rules of syntax. A simples formula may be just one symbol, for example $2$. More complicated formula are for example $2+x$, $x < y$ or $A^{-1}b$. If the formula contains no symbols representing variables, then the formula has a value; if it contains any symbols of variables, then the formula represents a function, and it has value only if we assign specific values to the variables. These values may be numerical like for the formulae $2+2$ or $\sqrt{4}$, but it may also be logical like for the formulae $2<3$ or $2=3$, or belong to some other category (it can be a set, vector, etc.).
An equation is a formula that has the specific form $$ formula1 = formula 2$$
where at least one of the formulae contains a variable, for example $Ax=b$. We call a specific value of that variable a solution of the exuation, if for that value of the variable the equation has a logical value 'true'. For example $A^{-1}b$ is a solution of equation $Ax=b$.
When we have a equation that simply says for example $x=A^{-1}b$ it can be said that the value of $x$ is given by the formula $A^{-1}b$, that is, this formual gives the only value of $x$ that solves the equation.
To sum up you can understand an expression $ x=A^{-1}b$ in two ways:

as a formula that for various values of $x4 may be true or false (an equation), or
as a statement that $A^{-1}b$ is the solution of some equation containing variable $x$.

In the cited fragment it is used in the second sense.

