What is an Independent Variable?

[Poweranimals]

What is an independent variable. Is that what you have to solve for?

Like for example in the problem; y^2 = x - y.. If x is the independent variable, is that what I have to solve for, or does it mean something completely unrelated?

 

[styler]

Yes, the idea is that y is dependent upon x or that we watch how y changes when we change x.
Or we think y depends on x in some way (given by an equation) as we allow x to vary over some set of numbers
We think of x as independent (like it takes on a certain set of values that we choose, not which are chosen by an equation) and y as dependent upon x (like the values of y are entirely determined by the equation(s) that realte(s) y to x.

Usually in an equation like y = (something with x)
we assume we know what values x can take like "all real numbers" or "all integers" or "all numbers between 0 and 1"

This is the same a sthinking of y as a function of x.

It can be confusing at first because if y varies as x varies then you could thin of y as independent and see how x changes correspondingly (this is what you would learn (partially) from "solving for x" in the equation
y = (something with x)
but since the general idea is that you know in advance what values x can take but not what values y might take we call x independent and y dependent.

 

[Zurtex]

What is an independent variable. Is that what you have to solve for?

Like for example in the problem; y^2 = x - y.. If x is the independent variable, is that what I have to solve for, or does it mean something completely unrelated?

Well as said before, generally speaking if x is the independent variable then it is put in the form y = f(x). However there is nothing wrong with your form, for example the standard equation of a circle is given as:

$$(x - a)^2 + (y - b)^2 = r^2$$

As apposed to the much more clumsy version:

$$y = \pm \sqrt{r^2 -(x-a)^2} + b$$

With your equation you could rearrange it like so:

$$y^2 = x - y$$

$$y^2 + y = x$$

$$\left(y + \frac{1}{2}\right)^2 = x + \frac{1}{4}$$

$$y = \pm \sqrt{x + \frac{1}{4}} - \frac{1}{2}$$

 

[gravenewworld]

Just think of an independent variable of a function as the the variable you can pick numbers for and plug it in. The dependent variable is the variable whose value depends on the value of the independent variable. For example y=x+2. X is the independent variable, you can pick any number and plug it in for x. Y is the dependent variable because Y's value depends on what ever number you choose for X.

 

[Gokul43201]

Except for reasons of application (or niceness of form), there is no intrinsic difference that makes one variable "independent". I think that kind of label is misleading and needs to be gotten rid of.

 

[Didd]

An independent variable means a variable whose values are not dependent.
for example; weight(w)=mass(m)*acceleration due to gravity(g).
w=m*g, here g is indpendent variable. In other way, it means a variable which is not affected.
You solve for that given variable if you need to solve for it. But the variable being independent or dependent is not a way to know which variable to solve for.
Have got an idea?

 

[styler]

Except for reasons of application (or niceness of form), there is no intrinsic difference that makes one variable "independent". I think that kind of label is misleading and needs to be gotten rid of.

I disdgaree.
I think the idea is that the range of the independet variable is determined in advance and that it defines the domain of the depenent varible so it makes sense to begin thinking in terms of realtions/functions.
there is of course the philosophical matter of bringing up the issue of whther x is considered to be "known" and y "unknown" which in principle could make a difference, in an empirical setting, for example.
But of course in maost cases how "x" realtes to "y" is no less fundamnetal than how "y" relates to "x".

But power animals don't worry about this point now.
For now just read what everybody wrote above and think about the variable for which YOU get to choose the numerical values as "independent" and the resulting values of the other variable that you get from aplying the equation as "dependent".

 

[HallsofIvy]

The distinction between "independent variable" and "dependent variable" makes sense only for functions. That is, if y= f(x) and a single value of x produces only one value of y, then x is the independent variable and y the dependent variable. Of course, if f is invertible then we could write x= f^-1(y) and say that y is the independent variable and x the dependent variable.

In the case of x²+ y²= 1, we can write neither x nor y as a function of the other variable so neither is "independent" nor "dependent".

In the original problem, y² = x - y, we can write x= y²+ y but cannot solve for a single value of y so we would say that y is the independent variable and x is the dependent variable.