regression with multiple independent variables vs multiple regressions with one independent variable For example, we want to use age and IQ to predict GPA.
Of course we can do a multiple linear regression, i.e. regress GPA on age and IQ.
My question is: can we do two simple regressions instead? First, regress GPA on age and discuss the relationship between GPA and age. Then, regress GPA on IQ and discuss the relationship between GPA and IQ.
I understand if IQ and age are uncorrelated, they are essentially the same. What if IQ and age are slightly correlated in practice? Which method is better? Fundamentally what's the difference between these two methods?
 A: To explain a little more. Multiple regression tests for the unique contribution of each predictor. So let's take your example and assume that IQ and age are correlated.
If you run a regression with IQ only the contribution of IQ can be visualized like this (red part):

But once you add age to the analysis, it looks something like that:

As you can see the unique contribution (red part) of IQ is smaller, hence beta for IQ will dicrease in this analysis.
I hope this makes it clear why both analysis answer different question: First analysis, using only IQ as the predictor, tells you how much IQ contributes to predict GPA in total, while in the second analysis you can see the unique contribution of IQ to explain variation in GPA apart from age.
Keep in mind, that this is a simple exmaple and there can be other things going on like moderation, mediation or suppression which can change your interpretation of the results.
A: You can do that. It answers a different question. 
If you include both independent variables then the results for each are controlling for the other. If you do them separately then they are not. 
A: What this would do is answer drastically different questions.


*

*Multiple regressions of one independent variable will give you an understand of 
the target variable varies with each output of each variable

*A regression with multiple independent variables would give you
coefficient estimates that let you know how the target variable
varies for a given change in the independent variable - controlling
for the other independent variables in the regression. 


In the first case you would not be taking into account the impact of certain factors such as wealth, gender, ... into account when looking at at the age coefficient on IQ. 
If for example, there is a disproportionate number of wealthy young people, that can have access to better education, better nutrients ... that will be implicitly absorbed in your "age" coefficient of your 1 independent regression variable. The regression might show that young people are "smarter", which might be true given your dataset, but the underlying factor might be attributable to wealth instead.
A: Your question says "Which method is better?". Better what for? If you want to predict GPA you might want to use both variables. If your question is about the relation between IQ and GPA, then you have no reason to add age to the Model. Hence, it depends on your research question what Model suits better. One point that appears unmentioned, is that not only beta but also the p values can change after addition of another predictor, leading to another interpretation of the results.
