# Basic Mathematics

Lets assume a protocol with a token $$T\_P$$. On an initial cycle is going to be paired with an USD-pegged stablecoin such as **USDC**. In that way, the LP is going to be $$\frac{T\_P}{USDC}$$, and the bonding curve formula given by Uniswap is:

$$x \cdot y = K$$

Hence,

$$T\_P \cdot USDC = K$$

This formula is responsible for calculating prices, and deciding how much $$T\_P$$ token would be received in exchange for a certain amount of USDC, or vice versa.

The formula states that $$K$$ is a constant no matter what the reserves (*x* or *y*) are. Every swap increases the reserve of either USDC or $$T\_P$$ token and decreases the reserve of the other.

$$(x + \Delta x) \cdot (y - \Delta y) = K$$

Where $$\Delta x$$ is the amount being provided by the user for sale, and $$\Delta y$$ is the amount the user is receiving from the DEX in exchange for $$\Delta x$$.

Since $$K$$ is a constant, we can asume that:

$$x \cdot y = (x + \Delta x) \cdot (y - \Delta y)$$

Before any swap is being made, we know the exact values of *x*, *y*, and $$\Delta x$$ (given by the input). In that way, we are interested in calculating $$\Delta y$$, which is the amount of USDC or $$T\_P$$ token the user will receive:

$$y - \Delta y = \frac{x \cdot y}{x + \Delta x}$$

$$-\Delta y = \frac{(y \cdot \Delta x)}{(x + \Delta x)} - y$$

$$-\Delta y = \frac{x \cdot y - y(x + \Delta x) }{x+ \Delta x}$$ 

$$-\Delta y = \frac{x \cdot y - y \cdot x - y \cdot \Delta x}{x+\Delta x}$$

$$-\Delta y = \frac{-y \cdot \Delta x}{x+ \Delta x}$$

Hence, after simplyfing the above equation to obtain $$\Delta y$$ , we get the following: 

$$\Delta y = \frac{(y \cdot \Delta x)}{(x + \Delta x)}$$