I am trying to learn simple linear regression and I understand everything up to the final step. After taking the two partial derivatives, you have this:
$\displaystyle \frac{\partial T}{\partial a} = \sum_{i = 1}^n \bigl [2a + 2bx_{i} - 2y_{i}\bigl]$
$\displaystyle \frac{\partial T}{\partial b} = \sum_{i = 1}^n \bigl[2ax_{i} + 2bx_{i}^{2} - 2x_{i}y_{i}\bigl]$
Then setting the equations to 0 yields this:
$\displaystyle \left \lgroup\sum_{i = 1}^n a \right \rgroup + b\left \lgroup \sum_{i = 1}^n x_{i} \right \rgroup - \left \lgroup\sum_{i = 1}^n y_{i} \right \rgroup = 0$
$\displaystyle a \left \lgroup \sum_{i = 1}^n x_{i} \right \rgroup + b\left \lgroup \sum_{i = 1}^n x_{i}^{2} \right \rgroup - \left \lgroup\sum_{i = 1}^n x_{i}y_{i} \right \rgroup = 0$
What happened to the coefficients of the terms? Why are they eliminated in that last step?