<s>[INST] Pattern matching puzzles Problems: I wish to have an equation for below data: 
$x = 2, y = 5$ 
$x = 2, y = 6 $
$x = 2, y = 7 $
$x = 2, y = 8 $
$x = 1, y = 9$ 
$x = 0, y = 10$ 
And Will you show me the steps to acquire that equation?
 [/INST] First sqap the values for $x$ and $y$. Then apply the Lagrange interpolation. The first 4 points lie on the the line $y=2$ Then do the Lagrange interpolation explained here:
After the application, the first 5 points then lie on the function
$$y = 2 - \frac{(x - 5) (x - 6) (x - 7) (x - 8)}{24}$$
Do this method again and the final function should be:
$$y = 2 - \frac{(x - 5) (x - 6) (x - 7) (x - 8)}{24} + \frac{(x - 5) (x - 6) (x - 7) (x - 8) (x - 9)}{40}$$
To get the function for your initial question, just swap $x$ and $y$. So the final function is: 
$$x = 2 - \frac{(y - 5) (y - 6) (y - 7) (y - 8)}{24} + \frac{(y - 5) (y - 6) (y - 7) (y - 8) (y - 9)}{40}$$
Or in expanded form:
$$x = -446+\frac{4894}{15} y - \frac{1123}{12}y^2+\frac{317}{24}y^3-\frac{11}{12}y^4 +\frac{1}{40}y^5$$

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