The arithmetic mean between two expressions is calculated by adding the expressions together and then dividing by 2.

Given the expressions $2x + 3y$ and $x - 5y$, the arithmetic mean is:

$\text{Arithmetic Mean} = \frac{(2x + 3y) + (x - 5y)}{2}$

Now, simplify the expression inside the parentheses:

$(2x + 3y) + (x - 5y) = 2x + x + 3y - 5y = 3x - 2y$

So, the arithmetic mean is:

$\frac{3x - 2y}{2}$

Thus, the arithmetic mean between $2x + 3y$ and $x - 5y$ is $\frac{3x - 2y}{2}$.

Would you like more details on this calculation or any related concepts? Here are some related questions:

1. How do you calculate the geometric mean between two algebraic expressions?
2. What is the difference between the arithmetic mean and the harmonic mean?
3. How do you find the arithmetic mean of three or more expressions?
4. Can the arithmetic mean be applied to complex numbers?
5. What is the significance of the arithmetic mean in statistics?

Tip: When simplifying algebraic expressions, combine like terms carefully to avoid errors.