Explaining the Vertical Line Feature in Graphing Equations

[Mark44]

x=y is an equation.
x+y is a formula.

In a typical equation, you put one expression on the left hand side, another expression on the right hand side and judge whether the values of those expressions are identical.

OK, I see what you meant now. What you're calling a formula I usually call an expression.

 

[Ned Eterminita]

The equation $x^2 = 4$ has two solutions. The equation $0x = 0$ has an infinite number of solutions. Neither of these is the formula for a function.

Thanks for clarifying that.

So I want to explain to my cousin why $0x = 0$ and $x = 0/0$ are not the same thing, in that x is undefined in the latter. How would you do it in a simple and concise manner?

 

[FactChecker]

Thanks for clarifying that.

So I want to explain to my cousin why $0x = 0$ and $x = 0/0$ are not the same thing, in that x is undefined in the latter. How would you do it in a simple and concise manner?

Exactly that way. He will have to deal with the fact that division by zero is not defined and get used to it.

 

[jbriggs444]

Thanks for clarifying that.

So I want to explain to my cousin why $0x = 0$ and $x = 0/0$ are not the same thing, in that x is undefined in the latter. How would you do it in a simple and concise manner?

x is not even undefined in the latter. That is because the latter is not even an equation. You cannot have a valid equation which contains an invalid term.

 

[Mr Indeterminate]

Right after my cousin takes my money and puts 1,000s of miles between him and I, does he reveal that he cheated…..

Look, I'm completely willing to accept that 0x=0 and x=0/0 are not the same thing, so long as a reason why this is the case is provided.

I reason the contrary because it's a matter of consistency, consider the below:

If ax=b and y=b/a then x=y

Now the above is correct when a=2 and b=6, as both x and y equal 3. It also correct when b=1 and a=0, as both x and y are undefined. Thus, it makes no sense for it to be incorrect when a=0 and b=0, as it's correct in all other instances.

Ok, now your turn….

 

[jbriggs444]

Right after my cousin takes my money and puts 1,000s of miles between him and I, does he reveal that he cheated…..

Look, I'm completely willing to accept that 0x=0 and x=0/0 are not the same thing, so long as a reason why this is the case is provided.

I reason the contrary because it's a matter of consistency, consider the below:

If ax=b and y=b/a then x=y

Now the above is correct when a=2 and b=6, as both x and y equal 3. It also correct when b=1 and a=0, as both x and y are undefined. Thus, it makes no sense for it to be incorrect when a=0 and b=0, as it's correct in all other instances.

To repeat what I've already said, it is improper to even write down x=0/0 except as an example of something that is improper. It is improper because, in the context of ordinary numeric division, the 0/0 does not mean anything. Writing down "x=0/0" is similar to writing down "x=$%3&^17". It is just gibberish.

By contrast, it is proper to write down 0x=0. The resulting equation does not constrain x. But that is ok. There is no rule that says that all equations must have solution sets that are singletons.

If one were to broaden the context to interval arithmetic (arithmetic on sets of real numbers) then one could define X/Y (for set X and set Y) as {z : zy = x for some y in Y and some x in X}. If this were done then {0}/{0} would denote the set of all real numbers. One might then be tempted to streamline the notation and say that $\frac{0}{0} = \mathbb{R}$. In this context, $\frac{0}{0}$ would be well defined and the solution set to 0x=b would be the same as the solution set to x=$\frac{0}{0}$.

But the context of interval arithmetic is not given. By default, we take "0" to denote a real number, "/" to denote ordinary real division and understand that division of one real number by another always yields a single definite real number except in the case of division by zero where the result is undefined and the operation may not be used.

 

[Mark44]

If ax=b and y=b/a then x=y

No, this conclusion is not correct. The second equation, y = b/a, is defined only if a ≠ 0.

Now the above is correct when a=2 and b=6, as both x and y equal 3. It also correct when b=1 and a=0

Sure, the conclusion (x = y) is correct when a = 2 and b = 6, but it is not correct when a = 0.
Division by 0 is not defined.

 

[Mark44]

Since the OP's question was answered, I am now closing this thread.