What does it mean to solve an equation?

Question

This question might be more philosophical than mathematical.

In school we are taught how to solve equations such as $x^2 - 1 = 0$ or $\sin(x) - 1= 0$. Solutions to these equations are quite simple. For example $x = 1$ and $x = -1$ are the solutions to the first equation. One could say that solving equation $f(x) = 0$ is same as finding the values of $x$ that satisfy the equation. To me this answer doesn't really tell what does it mean to solve a equation, because the meaning of the verb to find is ambiguous. If someone says that the solutions of the equation are all the numbers in the set $\{y \in \mathbb{R} | f(y) = 0\}$ has he or she solved the equation? I don't think he/she has.

Answer 1

Interesting question. I'd say that solving $f(x) = 0$ amounts to

• exhibiting the set S = $\{ x \mid f(x) = 0 \}$, typically by enumerating its members, or giving a sequence whose elements are all the members of $S$

• demonstrating that the listed or enumerated items are exactly equal to $S$.

So if I say that the solutions of $\sin x = 0$ are $n\pi, n = 0, \pm 1, \pm 2, \ldots$, I've given a purported solution set $S'$. I now need to show that for each element $t$ of $S'$, we actually have $\sin t = 0$, and that no other values of $t$ satisfy $\sin t = 0$.

The method by which I arrive at the set $S$ is not really germane, despite the active verb "solve"; the solution might come from algebraic or geometric manipulations, or it might come to me in a dream. But the second part -- the demonstration that the purported solution set is the actual solution set -- that must follow the rules of logic and mathematics.

This is, however, mostly opinion about common mathematical speech, rather than a fact about mathematics.

PS: For infinite solution sets that are not countable, Christian Blatter's answer starts to get at a good description, although it doesn't take into account things like "the solution set is all irrationals," where a parametrization of the set may be very hard to come up with. Roughly speaking, as the solution sets get more complicated, exhibiting the set gets more and more complicated. No big surprise there...

Answer 2

I agree with John Hughes; this is an interesting question. It is made more interesting by the observation that we often do answer questions like "Solve $f(x) = 0$" with something like $\{x \mid g(x) = 0\}$, which seems on the face of it like answering one question with another. Typically, of course, $g(x)$ is simpler in some sense than $f(x)$: For instance, we might answer

$$ \text{Solve for $x \in \mathbb{N}$ in }8x+1-y^2 = 0, y \in \mathbb{N} $$

with

$$ x \in \left\{\left.\frac{n(n+1)}{2} \,\,\right|\, n \in \mathbb{N}\right\} $$

where we might think of $x-n(n+1)/2$ as the equation $g(x)$ that characterizes the solutions to $f(x) = 8x+1-y^2 = 0$ for $y \in \mathbb{N}$. (There are better examples; this is just the one that comes to mind.)

What makes this characterization of $x$ simpler or better than the one that is posed as the problem? The explanation that I come up with is that we agree that—there is consensus that—the solution is a more immediately transparent description of the $x$ that solve the problem than the problem itself is. In other words, mathematics (like science in this regard) is a social activity, with (often unspoken) agreements about what constitutes progress toward a more primitive characterization of a mathematical object.

EDIT (2024-02-01): Sometimes, as in this case, at least part of the distinction can be made explicit like this: In the former case, the most straightforward and direct way to find all solutions is a loop that, for all you know, is unbounded; it's not clear there are always more solutions. In the latter case, the most straightforward and direct way is simple—simply run through all natural numbers and plug them into a finite expression. But it's not always that clear-cut.

When we begin, as students, we get used to the idea of solutions being concrete, like the number $4$ for $3x-12 = 0$. Later, we apprehend that mathematics is a big web or network of relationships, and solutions often merely move from an expression that is less simple or transparent to one that is more simple or transparent. What it is that simplicity or transparency actually represent is not usually made explicit, and I'm not sure that it can be made explicit in any universal way.

Answer 3

An equation, or a system of equations, defines a solution set $S$ as the set of all members $x$ belonging to some universe $X$ that satisfy certain conditions encoded in a formula, or a "story" ${\cal P}(x)$: $$S:=\{x\in X\>|\>{\cal P}(x)\}\ .\tag{1}$$ This is an implicit description of the set $S$. In most cases it is easy to check whether a proposed $x\in X$ actually belongs to $S$ or not.

Solving $(1)$ means producing an explicit description of $S$. Such an explicit description could consist in a proof that $S$ is in fact empty, it could consist in a finite list $S=\{x_1,\ldots, x_p\}$ of explicitly exhibited elements $x_k\in X$, or it could consist in a parametric representation $$f:\quad I\to X,\qquad \iota\mapsto x_\iota\in X\ ,\tag{2}$$ where $I$ is a certain "standard" set, e.g., $I={\mathbb N}$, $f(I)=S$, and $f$ is injective. In other words: Each element of $S$ is produced by $f$ exactly once in a well understood way.

Answer 4

IMO solving an equation is giving the solution set in extension ($S=\{-1,1\}$) rather than in intention ($S=\{x\in\mathbb R|x^2=1\}$).

The elements of the set can be specified via mathematical expressions, preferably that allow effective computation. This makes the solution constructive.

In cases that no mathematical expression is possible, one may consider the equation solved if root isolation has been performed, i.e. the enumeration of intervals guaranteed to contain exactly one root, potentially allowing computation by means of numerical methods.

For instance, I consider that stating "the equation

$$x^7=x+1$$ has a single real root" is sufficient as a solution. Adding "the solution lies in $(1,2)$" is better but is just a hint.

In a nutshell, I see the solution of an equation as the discussion of root separation. In the end, you can refer to every root unambiguously (even without knowing its exact value).

The equation $x^2=1$ has one negative root, $x_-$ and one positive root, $x_+$. Without knowing them, we can anyway affirm $x_-=-x_+$.

Answer 5

If someone says that the solutions of the equation are all the numbers in the set {y∈R|f(y)=0}{y∈R|f(y)=0} has he or she solved the equation? I don't think he/she has.

i think you have to understand 'solve' more like simplify in this context. From an arbitrary equation to the solution is like from abstract to specific. You specifie the abstract rules of your model. You have an equation, which represents something in abstract form, you solve it, you have the specific objects.

Answer 6

A prerequisite to be able to answer the question "What does it mean to solve an equation?" is to have a definition of the word "equation".

A definition is: an equation is an equality containing one or more unknown(s). Do you agree with this definition ?

If yes, solving the equation consists either :

• of determining which values, or particular form(s), of the unknown(s) make the equality true,

• or proving that no value and no particular form of the unknown(s) make the equality true.

Answer 7

First of all let's look for what solution means:

The Free Dictionary: To find an answer to, explanation for, or dealing with (a problem, for example).

So.

We have a result on the right side of the equal sign.

If you see there a number then the word solution doesn't make a sense. It is ambitious.

But.

If you see there a result of either an event or events, we can say that:

Wow! Something happened and we got 2, an accident, and so on.

Then we start to find the possible causes to explain why the result is there.

I think the verb, solve, is the best describer of whole finding causes process.

Answer 8

Your example of {$y \in \mathbb R | f(y) = 0$} as solution for the equation $f(x)=0$. It's simply stating that all solutions for f(x) are all solutions for f(x). A more descriptive solution set would be {$y \in \mathbb R | g(y)=0$}.

And now we solve the solutions recursively. You look at the solution set of $g(x)$ and then at the function which describes it and so on, until you reach a trivial solution, which is a list of numbers, which can be not simplified*, e.g. {$2n | n \in \mathbb N$}. This set can be written down without solving on equation.

*With simplified I mean that there is no function in the solution set anymore and thus does not require anymore recursion.