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Understanding the distributive property is essential when simplifying algebraic expressions. It involves the multiplication of a number or variable by each individual addend within a set of parentheses. Simply put, it allows for spreading out multiplication across addition or subtraction.

In the given exercise, we apply the distributive property to two terms: \(7(3 y-5)\) and \(2(4 y+3)\). Here's how it works: for the first term \(7(3 y-5)\), the number 7 is distributed to both 3y and -5, leading to \(7*3y\) and \(7*(-5)\). Similarly, in the second term \(2(4 y+3)\), the number 2 is distributed to 4y and +3, yielding \(2*4y\) and \(2*3\).

The property is typically expressed as \(a(b + c) = ab + ac\), demonstrating that the distributive property allows us to break apart equations for easier manipulation. After distribution, our original expression becomes \(21y - 35 + 8y + 6\), which sets the stage for further simplification through combining like terms.

Once we've used the distributive property to expand an algebraic expression, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power, which means they can be added or subtracted from each other just like regular numbers.

In our problem, after distribution, we identified the like terms involving the variable y: \(21y\) and \(8y\). Combining them, we simply add their coefficients (the numbers in front of the y) to get \(21y + 8y = 29y\). This is because the variable part (y in this case) does not change.

We then do the same with the constant terms, which are \( -35\) and \(+6\). Adding them together, we have \( -35 + 6 = -29\). The key to combining like terms is to keep an eye on the signs preceding the terms, as they determine whether you add or subtract the coefficients. After combining, the expression simplifies to \(29y - 29\), which is a much more straightforward representation of the original problem.

Algebraic manipulation refers to the various methods used to transform complex expressions into simpler or alternative forms without changing their value. This encompasses both the distributive property and combining like terms, as well as operations such as factoring, expanding, and simplifying expressions.

It is important to know not just the steps but why we perform them. By distributing multiplication over addition, we can simplify expressions within parentheses, which lays the groundwork for further reduction. Then, by combining like terms, we consolidate the expression into as few terms as possible, making it easier to understand and, if needed, solve for the variable.

Mathematicians developed these techniques for simplification to handle more complex algebraic problems, including solving equations and inequalities. The ability to manipulate algebraic expressions fluently is foundational for higher-level mathematics and is immensely useful in problem-solving situations across various disciplines.