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I'm trying to pin down the difference between "unknown" and "variable". I have always understood that in the equations $2x+1=10$ or $x^2+5x+6=0$, $x$ is an unknown (short for "unknown constant"), since its value can be determined. In the expression $2x+1$, however, $x$ can take any value, therefore it is a variable.

What about in the equation $2x+3y=10$? $x$ and $y$ can both take infinitely many values, but once one is fixed, the other becomes fixed. Does this mean they are both variables? Does it mean that one (say $x$) is a variable, but the other is an unknown (since its value is determined by the variable)?

I'd appreciate some insight. Thanks.

zoli
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Trying to get better
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    "unknown" and "variable" are no formal terms. They are mostly used interchangeably. Which one is more appropriate can often be seen not only from the formula, but a problem statement is necessary. So saying this, $x$ and $y$ in $2x+3y=10$ can be considered both, as long as you do not state where this formula comes from or what you are using it for. – M. Winter Oct 16 '17 at 12:09
  • Regarding "variable", see the post: logic-in-the-use-of-variables. – Mauro ALLEGRANZA Oct 16 '17 at 12:32
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    See Equation: "In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation." – Mauro ALLEGRANZA Oct 16 '17 at 12:41
  • Does this answer your question? A formal definition of a variable. – Jam Oct 15 '22 at 08:57

4 Answers4

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It depends what your question is. While they are kind of interchangeable, these two terms are used in different contexts. Unknown is usually employed in equations,so for example you could ask how to solve the equation $2x=1$,where $x$ is the unknown. On the other hand the term variable is more used in case of functions. You could ask for example what is the second derivative according to the $x$ variable of the function $f(x,y)=x^2+y+2$. Hope that clears things up a bit.

RGS
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Keen
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  • Well, I'm happy with that difference already. It's the specific case of an equation with more than one variable/unknown where I think the distinction is unclear. – Trying to get better Oct 19 '17 at 12:43
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The term variable is used in two contexts. In both instances the variable can take on any value from a chosen set.

The first is in identities: $4x \equiv x+x+x+x$. Notice that the choice of $x$ makes no difference to the truth value of the identity, hence it is free to vary.

The second is in functions: $f(x,y) \equiv x+y$. Notice again that the equivalence is true for all values of $x$ and $y$.

In contrast, if we use an equation or inequality that acts like a condition on that variable such as $2x = 5$ then the variables becomes an unknown since it takes on specific values.

It becomes even more fun when we consider formulae such as $F=ma$. One could say that we have two variables and one unknown since we have a free choice for two of them and the 3rd is then forced to be an unknown.

If we have simultaneous equations such as $a+b=6$ and $a-b=4$ then we have two unknowns and no variables.

It ultimately comes down to how restricted your variable space is. The more restrictions, the less variables you have.

Ben Crossley
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Variable is opposed to constant. Unknown is opposed to known.

There can be both (1) unknown values of a variable, and (2) unknown constants.

A variable is not an unknown per se; only some value of a variable, satisfying a given condition.

Example: in $ f(x)= 3x+2$ , $x$ is not an unknown. It would not make sense to ask the question : " what number is hidden behind $x$ in $f(x)= 3x+2$? ". $x$ becomes an unknown when you ask the question, say : " for which value of $x$ is $f(x)= 10$ ? ".

An unknown is not per se a variable, a constant can also be an unknown.

Example: What are the coefficients $a, b, c$ of the parabola $ax^2 + bx+c =0$ that opens upward and that passes through the points $(0,0) , (-3,3) , (3,3)$. These coefficients are constants , not variables; but they are the unknowns of the problem.

user1147844
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Vince Vickler
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    To add to this answer: The terminology for these placeholders is informal, and usage is guided by context and framing or intended signalling; here is an outline that is consistent with Vince's answer. – ryang Mar 03 '23 at 07:03
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I would say...a variable is an unknown but an unknown doesn't necessarily have to be a variable. A variable means it could be any number, it is not fixed but a unknown means it is a specific number that we do not know as yet. Therefore a variable is an unknown because it could be any number but an unknown doesn't have to be a variable because it is a fixed number that we do not know. Hope that makes sense.

D. Sterling
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