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Understanding how to properly apply the distributive property is akin to mastering a fundamental building block in algebra. When faced with an expression such as 5(3y - 2), the distributive property allows us to multiply the term outside the parentheses by each term inside. Mathematically, this is expressed as a(b + c) = ab + ac. In the case of our example, you distribute the 5 to both 3y and -2, resulting in two separate products: 5 times 3y, and 5 times -2, which simplify to 15y and -10, respectively.

Furthermore, when subtracting a grouped expression like -(7y + 2), it's crucial to distribute the negative sign too, turning the subtraction into addition of a negative. Basically, this is the process of removing parentheses: a - (b + c) = a - b - c. Applying this, we adjust both terms within the parentheses to their negatives, giving us -7y and -2. This step sets the stage for the next phase in simplification, which is combining like terms.

Once we've distributed our terms correctly, the pathway to simplification requires us to combine like terms. Like terms are composed of the same variable raised to the exact same power and can be combined through addition or subtraction. In the distributed form of our exercise, 15y - 10 - 7y - 2, the like terms are the 'y' terms (15y and -7y) and the constant terms (-10 and -2).

To combine them, simply add or subtract the coefficients (the numerical parts) of the like terms. For the variable y, subtract 7y from 15y to obtain 8y. For the constants, add -2 to -10 resulting in -12. The concept is straightforward: when variables and exponents match, merge them to simplify your algebraic expression.

The final step in our problem-solving journey is algebraic manipulation. This essentially involves utilizing various algebraic techniques to transform complex or abstract expressions into simpler and more solvable forms. After distributing and combining like terms, our example simplifies to a neat 8y - 12.

Algebraic manipulation doesn't stop here; it encompasses operations such as factoring, canceling, expanding, and rearranging terms. Mastery of these techniques provides the toolkit needed to tackle a wide array of algebra problems, from the most rudimentary to the profoundly complex. While the provided exercise is already in its simplest form, ongoing practice and a solid grasp of algebraic manipulation principles can equip students to address a broader spectrum of problems with confidence and agility.