I'm trying to understand how the formula for Lagrange Interpolating Polynomials comes about by looking at the basic case of Linear Lagrange Interpolating Polynomials. I found this derivation but it gives me no understanding as to how the final line comes about. Sure I can follow all the steps, but I would be very unlikely to make this discovery myself. Can some one please explain the intuition behind this approach? Please note I haven't get to the point where I can make much sense of this answer: https://math.stackexchange.com/a/523922/442515
\begin{align} \quad y - y_1 = \frac{y_1 - y_0}{x_1 - x_0} (x - x_1) \\ \quad y = y_1 + \frac{y_1 - y_0}{x_1 - x_0} (x - x_1) \\ \quad y = \frac{y_1(x_1 - x_0) + (y_1 - y_0)(x - x_1)}{x_1 - x_0} \\ \quad y = \frac{y_1x_1 - y_1x_0 + y_1x - y_1x_1 - y_0x + y_0x_1}{x_1 - x_0} \\ \quad y = \frac{- y_1x_0 + y_1x - y_0x + y_0x_1}{x_1 - x_0} \\ \quad y = \frac{y_1(x - x_0) + y_0(x_1 - x)}{x_1 - x_0} \\ \quad y = y_0 \frac{(x_1 - x)}{x_1 - x_0} + y_1 \frac{(x - x_0)}{(x_1 - x_0)} \\ \quad y = y_0 \left ( \frac{x - x_1}{x_0 - x_1}\right ) + y_1 \left ( \frac{x - x_0}{x_1 - x_0} \right ) \end{align}