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I tried to find the estimators of $\hat{\beta_1}$ and $\hat{\beta_0}$ via the least-squares method algebraically. Somehow I seem to have messed up. Can you tell me where?

My Calculations.

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Your calculations seem fine, you should only continue rearranging the terms and using the fact that from the basic definition of a mean, we have:

$$ \sum x_i = n\bar x = \sum \bar x .$$

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