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A variable does not usually "cause" another. A change in one variable may result in a change in another. Such a change may be systematic (the relationship between the two variables is precise) or it may be statistical or correlational.
The reasons for either type will often depend on disciplines of knowledge other than mathematics, such as physics or economics etc.
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How do you know whether a relation is a function?
Use the definition of a function. If, for any value of one variable, there is only a single possible value of the second variable, then the second variable is a function of the first variable. The second variable is often called the "dependent variable". If you can solve an equation explicitly for the dependent variable, then it is a function. If you can NOT solve it for a variable, it may or may not be a function - it turns out that some equations are hard or impossible to solve explicitly for one of the variables.
How to identify independent and dependent variables in an experiment?
If changes in one variable do not affect the outcome of another variable, then the second variable is independent. A variable that is not independent is dependent.
Why is it important to be able to solve for the variable in a formula?
Solving for one variable makes it easy to put in a value for the other variables, and find a value for the first variable.
When an experiment shows that one variable makes another variable change and no other variables have any effect the experiment shows what?
causation
Which variable is independent and which variable is dependent?
Oh honey, the independent variable is the one you can control and manipulate, like a puppet master pulling the strings. The dependent variable is the one that sits back and gets affected by the independent variable's shenanigans, like a poor unsuspecting victim. So, in simpler terms, the independent variable is the cause, and the dependent variable is the effect.
