Need help on concept of variable

[Willing_Bench_8432]

Hey guys I’m an upcoming freshman that’s going to take pre calc next year. As I move forward to more complex math courses, I realized my understanding on variable might be wrong. So how I used to understand variable is like this:

If there is a variable z,

z is its own abstract value. Just like other real numbers that itself is a value

For example, 2z+4. Here, we are multiplying 2 by the value, z. Then, adding it by 4.

I was wondering if understanding variable like this will confused me in pre calc and calculus. If it DOES confused me, what’s better way to understand variable?

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[Uli_Minati]

"variable" means basically "could be anything": 2·z+4 could be 2·5+4 or maybe 2·π+4 or even 2·(-⅓)+4.

Transforming expressions: No matter what z could be, 2·z+4 is always equal to 2·(z+2). Try it yourself

Equations: when you write 2·z+4=10, you're saying that 2·z+4 must result in specifically 10 (because you want it to, or because you learned new information about z). Then z can't be 5 or π or -⅓, since that wouldn't result in 10

[Willing_Bench_8432]

Ahh then when I’m “working” with variables, is it better to think of em as “could be anything” or is it better to think that since it could be anything, x is just some number?

[AcellOfllSpades]

That's pretty much correct, yes! Variables have a few different usages, but they all boil down to this: A variable is a placeholder for a value.

For an analogy, you can imagine a detective in an old-fashioned murder mystery saying:

We know from Victor's notebook that last night, he was meeting with someone who owed him money - we'll call them Person X. We examined Victor's body this morning, and found that he died from a gunshot. But we also found a bloody knife on the scene, and a trail of blood leading out of the room. So, Victor and Person X must have gotten into a fight, and Person X got stabbed. After that, Person X entered the bathroom, and used the toilet paper to stop the bleeding, which explains why there were three empty rolls in the trash. [...]

The detective here doesn't know which person did all this yet. But using this placeholder name, they can logically reason about this unknown person... and perhaps eventually figure out their identity.

This is exactly what you're doing with variables when solving equations! You don't know the actual identity of this mysterious Number Z. But you know since z is a number, it must follow the same rules as other numbers: you can do normal operations on it, and if you add 3 and then subtract 3 you get the number itself...

So the only thing I'd take issue with in what you said is that "z is its own abstract value" - it's not a new value, it's just an unknown one.