Meaning(s) of the Word "Equation"

Question

According to the dictionary, an equation is "a statement that the values of two mathematical expressions are equal."

More specifically, can the word "equation" properly denote a particular formulation of two equal expressions, the more abstract notion represented by that formulation and all equivalent ones, or both? For example, when asked to "place the equation x = .5y in slope-intercept form," are "x = .5y" and "y = 2x" really two forms of the same abstract notion of an equation, are they two different equations that merely share their solutions, or is "equation" defined in both ways?

The word "statement" in the definition might seem to indicate a requirement of being formulated in a particular way, but I'm suspicious that this would press the definition for more technical precision than was intended. If "equation" does turn out to be more or less synonymous with "equation form," what word can be used to describe the more abstract notion that mathematically equivalent equation forms are meant to reflect?

Answer 1

Mathematician here. You are correct to say that the following are both equations themselves:

• y=2x
• x=0.5y

But you also seem to be asking about the relationship between those individual statements. This is the notion of logical equivalence. We mathematicians would say those two statements are logically equivalent because, although they may look different, they assert the same underlying information.

The double-arrow symbol "⇔" is often used to indicate this equivalence:

y=2x ⇔ x=0.5y

The above statement indicates two assertions:

1. If "y=2x" is true, then "x=0.5y" must necessarily be true, as well.
2. And vice-versa: if "x=0.5y" is true, then "y=2x" must also be true as well.

In a way, though, you can interpret the symbol "⇔" in the above logical statement as fulfilling the same role as "=" does in the equation "y=2x". That equation is an assertion that two "things" are identical, and those things happen to be numerical values. Similarly, the logical statement above is an assertion that two "things" are identical, and those things happen to be equations. So, we do have specific terms (equation, logical equivalence) for these two kinds of statements because they apply in different contexts, but it is possible to view them as quite similar notions (although this is getting more into the philosophy of mathematics than English language usage).

Answer 2

Both the "x=0.5y" and "y=2x" are equations by the definition you proffered. Each one comprises two mathematical expressions (e.g. "x" and "0.5y") and the assertion is that these two expressions are equivalent (hence the "="). The "statement" is that you utter, write, or otherwise express the information conveyed in the equation.

Whether the two equations are equivalent is not really a question of language but of mathematics or philosophy. If you are looking for something more "abstract" perhaps you should consider asking http://math.stackexchange.com or http://philosophy.stackexchange.com?