Why $b_0$ and $b_1$ from linear regression $(y = b_0 + b_1\cdot x)$ are calculated like follows? How is the formula for such variables derived?

Question

I can't understand the logic of the following formula $$b_1 = \displaystyle\frac{\sum(x-\overline{x})\cdot(y-\overline{y})}{\sum(x-\overline{x})^2 }$$ where $\overline{x}$ and $\overline{y}$ are arithmetic means of $x$ and $y$, respectively. And if we calculated $b_1$, why do we choose next formula for calculating $b_0$? $$\overline{y} = b_0 + b_1\cdot\overline{x}$$

Answer 1

These values for the slope and the y-intercept of the regression line are found by the least square method.

They find the line for which the sum of the squared errors are minimized.

The method requires taking partial derivatives of the sum squared errors and solve for those constants by making the partial derivatives equal $0.$