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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}

Robin Andrews
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    Don't look at Lagrange polynomial as derivation from basic line equation. Look at it from the intuition that the polynomial is forced to pass through given points, as the answer you linked suggests. – utkarshk5 Jun 02 '21 at 15:55

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