why MRS xy is equal to -Px / Py ? and not equal to Px / Py ? why the minus sign ?

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Solution

Marginal rate of substitution means the rate at which a consumer sacrifices one commodity to have an extra unit of another commodity, keeping his utility constant. It can easily be shown by taking ratio of the prices of two commodities.
It also shows the slope of the indifference curve.
Mathematically
$\frac{- d y}{d x} = \frac{P x}{P y} o r \frac{d y}{d x} = \frac{- P x}{P y}$
Since the indifference curve is negatively sloped and the MRS goes on decreasing, we put up a negative sign before the ratio of prices.

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