What is the difference between equation and formula?

Question

Sometimes equation and formula are used interchangeably, but I was wondering if there is a difference.

For example, suppose we can calculate a car's fuel efficiency as:

mpg = distance traveled in miles / the fuel used in a gallon

Is that an equation or formula?

Answer 1

An equation is meant to be solved, that is, there are some unknowns. A formula is meant to be evaluated, that is, you replace all variables in it with values and get the value of the formula.

Your example is a formula for mpg. But it can become an equation if mpg and one of the other value is given and the remaining value is sought.

Answer 2

I'd say an equation is anything with an equals sign in it; a formula is an equation of the form $A={\rm\ stuff}$ where $A$ does not appear among the stuff on the right side.

Answer 3

Please down vote me if you wish - but I would say these words are really synonyms to each other. They both express that there is some underlying relation between some mathematical expressions.

Answer 4

An equation is any expression with an equals sign, so your example is by definition an equation. Equations appear frequently in mathematics because mathematicians love to use equal signs.

A formula is a set of instructions for creating a desired result. Non-mathematical examples include such things as chemical formulas (two H and one O make H2O), or the formula for Coca-Cola (which is just a list of ingredients). You can argue that these examples are not equations, in the sense that hydrogen and oxygen are not "equal" to water, yet you can use them to make water.

Mathematicians have long since realized that when it comes to numbers, certain formulas can be expressed most succinctly as equations. For example, the Pythagorean Theorem $a^2+b^2=c^2$ can be thought of as a formula for finding the length of the side of a right triangle, but it turns out that such a length is always equal to a combination of the other two lengths, so we can express the formula as an equation. The key idea is that the equation captures not just the ingredients of the formula, but also the relationship between the different ingredients.

In your case, "mpg = distance/gallons" is best understood as "a formula in the form of an equation", which means that in this instance the two words are interchangeable.

Answer 5

TL;DR I'd say it really depends on the context.

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What I remember in high-school/secondary school:

We were given problems like

Given length and area of a rectangle, find its width.

(Not exactly rectangle. that's of course more primary/grade school. Can be rectangular prism or whatever.)

The 'formula' is $A =wl$.

The 'equation' is what you get when you plug in the given values for $A$ and $l$. So if you have $A=10$ and $l=7$, then the equation is $10=7w$.

At the time, I thought it was very nit-picky/subjective/conventional/contextual. Now, I still do but I have the 10,000+ rep and bachelor's and master's degrees to complain about it.

According to my secondary school teachers, what you provided is a formula, but that's in the context of filling in the blanks of equation and formula in school.

Conclusion: I'd say it really depends on the context. If you define equation as a statement with an equal sign, then every formula with an equal sign is an equation... By the way, it seems on Wikipedia that there are no inequalities that are formulas.

Answer 6

An equation is a statement that connects two expressions with an $=$ sign and asserts their equality.

An identity is an equation that holds for every variable tuple for which the equation is defined:

• $(x+y)^2\equiv x^2+y^2+2xy$

A conditional equation holds for some variable tuple(s):

• $x^2+ky^2=1\quad$ (the parameter $k$ is an arbitrary constant, varying to generate a family of equations)
• $2x^2+3x-5=0\quad$ (in the context of equation-solving, $x$ is an unknown)
• In a formula (the rule of a function) like $$V=\pi r^2h,$$ each input tuple returns an output called the subject.

An inconsistent equation holds for no variable tuple:

• $|2x|=x-1$

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Addendum

A constant whose value is unspecified, a parameter and an arbitrary constant all mean the same, and each has a unspecified value (unless instantiated) in the context or problem. The first term emphasises the placeholder's fixed value within the context (in contrast with a variable), the second term emphasises its varying value across contexts (in similarity with a variable), while the third term emphasises that its choice of value isn't important.

In the context of a constraint problem, a variable or a parameter or lettered constant can also be called an unknown.

Answer 7

In a formula all the variables can be arbitrarily chosen. An equation admits only particular values of non-constant variable/s.