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Understanding how to solve algebraic equations is fundamental in mathematics. An algebraic equation states two expressions are equal, and typically involves variables which represent unknown quantities. The process of finding the values of these variables that make the equation true is called 'solving the equation.'

For example, consider the equation used to find the inverse of the function in our exercise: \( x + 6 = 5y \). The objective is to isolate the variable we want to solve for, in this case, \( y \). By performing inverse operations, we reverse the actions applied to \( y \) to get \( y \) by itself on one side of the equation. Here, we first add 6 to both sides to balance the equation, and then divide by 5 to isolate \( y \). These steps result in \( y = \frac{x+6}{5} \), which is the inverse function we are looking for.

The concept of function transformation refers to changing a function's graph or output values. It's a powerful way to understand complex functions by adjusting simpler ones. When finding inverse functions, we essentially perform a very specific transformation: we 'flip' the function over the line \( y = x \).

To illustrate this, let's look at our original function \( y = 2(4x - 3) - 3x \). Finding its inverse involves swapping \( x \) and \( y \) to get the new function \( x = 5y - 6 \). This flipping over the line \( y = x \) leads to the transformation that reveals the inverse. Understand that this transformation doesn't change the function's input-output pairs but rather represents them in a different way, indicating an inverse relationship between the variables.

The distributive property is one of the most commonly used properties in algebra. It allows you to multiply a single term by each term within a parenthesis and add the products together. Mathematically, it's expressed as \( a(b + c) = ab + ac \).

In our step-by-step solution, the distributive property is employed in the first step, where we multiply 2 by each term inside the parentheses: \( y = 2(4x - 3) \). Applying the property, we get \( y = 2 \times 4x + 2 \times (-3) \), which simplifies to \( y = 8x - 6 \). Recognizing and applying the distributive property correctly is critical when simplifying expressions and solving equations, and lays the groundwork for more advanced mathematical concepts.