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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.

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  • $\begingroup$ Do you mean this question math.stackexchange.com/questions/38155/… ? $\endgroup$
    – lhf
    Commented Jul 18, 2019 at 11:30
  • $\begingroup$ Actually, I'd rather say that the formula $A^{-1}b$ can be used to compute $x$ such that the equation $Ax=b$ holds. $\endgroup$ Commented Jul 18, 2019 at 11:38
  • $\begingroup$ @lhf Yes. you link and my link are the same one $\endgroup$
    – JJJohn
    Commented Jul 18, 2019 at 11:39
  • $\begingroup$ @uniquesolution I've explained the difference between my question and your link. would you please point out a more specific link or "contents" of any answer in that post to provide a concrete case that is a formula rather than an equation, in the context of mathematics $\endgroup$
    – JJJohn
    Commented Jul 18, 2019 at 11:47
  • $\begingroup$ I'm voting to close this question as a duplicate, because the question itself does not demonstrate its difference with the question linked by @uniquesolution, and the answers it attracts demonstrate the similarity. The burden is on the OP to post a concrete question that is demonstrably different than the earlier question. $\endgroup$
    – Lee Mosher
    Commented Jul 18, 2019 at 13:01

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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.

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  • $\begingroup$ Thanks for your answer. Would you please provide a concrete case that is a formula rather than an equation, in the context of mathematics. $\endgroup$
    – JJJohn
    Commented Jul 18, 2019 at 11:42
  • $\begingroup$ @fuDL You're getting it wrong still. Every equation is a formula. But if you mean to ask for a formula that's not an equation, then any expression (well-formed or not) not containing the symbol $=$ or anything equivalent would do. However, I'll give an example you should know. So say $$x+y,$$ which instructs you to do a single thing -- to form the sum of $x$ and $y.$ $\endgroup$
    – Allawonder
    Commented Jul 18, 2019 at 13:19
  • $\begingroup$ Thanks you, you just gave what I want. What do you mean "getting it wrong still"? I was asking for a formula that's not an equation by saying "a concrete case that is a formula rather than an equation", what is wrong with the expression? $\endgroup$
    – JJJohn
    Commented Jul 18, 2019 at 13:38
  • $\begingroup$ @fuDL Glad you finally got what you wanted. But when you said rather than, you were opposing two things in different classes, whereas as I explained the classes in question are not mutually exclusive. (But now I gather you didn't intend this.) $\endgroup$
    – Allawonder
    Commented Jul 18, 2019 at 15:43
  • $\begingroup$ Can you provide any citation or reference which makes the distinction which you are making? I have never seen a "formula" formally (or even informally) defined in this manner. Moreover, your comments directly contradict the answer given by Magma, which asserts that every formula is an equation, but not the converse (you say that every equation is a formula, but not the converse). $\endgroup$
    – Xander Henderson
    Commented Jul 18, 2019 at 16:49
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A formula is any expression built from mathematical symbols built in accordance to their rules of syntax. A simple 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 $$ formula \; 1 = formula \; 2$$ 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 equation, 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 formula 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 $x$ 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.

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    $\begingroup$ "where at least one of the formulae contains a variable": No, for example $2+2=4$ is an equation; for that matter so is $0=0$. $\endgroup$ Commented Jul 18, 2019 at 14:22
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    $\begingroup$ @DavidC.Ullrich I wouldn't call 2+2=4 or 0=0 equations. They are equalities. As I understand it, equations require one or more variables in respect to which they may be potentially solved. $\endgroup$ Commented Jul 18, 2019 at 14:37
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    $\begingroup$ I can see that's your understanding. That's why I didn't claim that you would call $2+2=4$ an equation. It is an equation, regardless of your understanding, regardless of whether you'd call it an equation. $\endgroup$ Commented Jul 18, 2019 at 14:53
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    $\begingroup$ @DavidC.Ullrich Nothing is or isn't an equation on its own merit, everything depends on how we define what equation is, that is what objects we decide to call "equations". There's no such thing as a correct or incorrect definiton. You're free to prefer a different definiton than mine. $\endgroup$ Commented Jul 18, 2019 at 15:02
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    $\begingroup$ Yes, definitions are arbitrary; it doesn't follow that standard definitions don't exist. If I say "If $f(t)=\sin(t)$ then $f'=f$" that's _wrong; saying "My definition is $\sin(t)=\sum t^n/n!$; you're free to prefer a different definition" doesn't change that. $\endgroup$ Commented Jul 18, 2019 at 15:11
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A formula is a special kind of equation, namely one that is solved for the unknown.

For example, in its general form the equation $a^2+b^2 = c^2$ in the Pythagorean Theorem is just that, an equation.

But if you rewrite it as $c = \sqrt{a^2+b^2}$, then you have a formula to get the hypotenuse of a right-angled triangle from its legs.

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  • $\begingroup$ Thanks for your answer. Would you please provide a concrete case that is a formula rather than an equation, in the context of mathematics. $\endgroup$
    – JJJohn
    Commented Jul 18, 2019 at 11:42
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    $\begingroup$ I'd say within the context of mathematics it doesn't get more concrete than the example I already provided. $\endgroup$
    – Magma
    Commented Jul 18, 2019 at 12:29
  • $\begingroup$ Can you provide any citation or reference which makes the distinction which you are making? I have never seen a "formula" formally (or even informally) defined in this manner. $\endgroup$
    – Xander Henderson
    Commented Jul 18, 2019 at 16:46

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