Why is it often said that dependent variable depends on the values of independent variable

Question

I hear it often from my teachers that dependent variable is called "dependent" because it depends on the values of independent variable.

But I think, you can say the same for independent variable, how is independent variable any different then?

Eg : $y = 5x$

• $y$ is dependent variable.

• $x$ is independent variable.

If I put $x= 5$, then $y= 25$. Surely I can also say, if $y = 25$ then $x$ has to be $5$.

Don't they kinda depend on each other? Then what's the point of calling one dependent and other independent? Why do we even call them that in the first place?

Answer 1

Indeed in this case they do depend on each other. The particular function is invertible: knowing $x$ you can find $y$ and vice versa. So sometimes you can think of either variable as the "independent one".

If you are driving at $30$ miles per hour and you want to know how far you can get in three hours you are thinking of time as the independent variable and distance as dependent: if you know how much time you take you can figure out how far you go.

You could equally well think of the same relationship the other way around. If you need to travel a certain distance you can figure out how long it will take. The time depends on the distance.

When the variables are "$x$" and "$y$" it's traditional to think of $x$ as the independent variable. In your example you can solve for $x$ in terms of $y$ and think of $y$ as independent. If the relation happened to be $y = x^2$ or $y = \sin x$ you couldn't do that.

Answer 2

There is nothing mathematically objective about dependent and independent variables. It's all about interpretation. In an applied setting (for instance, $x$ is an amount of something you buy and $y$ is the resulting cost, or $x$ is the time you spend doing something and $y$ is the amount you finish) the independent variable is often the one you have direct control over, while the dependent variable is the result.