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When working with algebraic expressions, the distribution of numbers is a crucial process to simplify and expand expressions. It involves the use of the distributive property, which can be articulated as 'a(b + c) = ab + ac'. This means you multiply a single term by each term inside a parenthesis.

Using the exercise provided, with the expression \(5(3 y-2)-(7 y+2)\), we start by applying this property. We distribute the 5 across the \(3 y-2\), translating to \(5 \times 3 y\) and \(5 \times -2\), which simplifies to \(15 y - 10\). Notice how the operation inside the parenthesis dictates how the number outside is distributed—multiplication for both terms and a sign change for the negative.

To remember this, it's helpful to think of the distributive property as dispersing a common factor across terms within parentheses, effectively removing the parentheses and creating an expanded form of the expression.

Combining like terms is another vital step in simplifying algebraic expressions, a technique that allows you to merge terms that have the exact same variables raised to the same power. The key aspect of this step is recognizing which terms are like terms. For instance, terms with \( y \) can be combined, but they cannot be combined with constant terms

Take the result after distribution from our example above, \(15 y - 7 y - 10 - 2\), where \(15 y\) and \( -7 y\) are like terms, as are the constants \( -10\) and \( -2\). To combine them, you simply add or subtract their coefficients: \(15 - 7\) for the \( y \) terms, giving \(8 y\), and \( -10 -2\) for the constants, resulting in \( -12\).

Algebraic operations include addition, subtraction, multiplication, and sometimes division, which we use to simplify or manipulate algebraic expressions. The correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides you through which steps to take first.

In our problem, after distributing the numbers and combining like terms, we perform the remaining algebraic operations, which are addition and subtraction in this case. We add the coefficients of the \( y \) terms together and sum up the constants, leading to our final simplified expression \(8 y - 12\).

It’s essential to perform these operations correctly and in the right sequence to ensure the expression is correctly simplified. In doing so, you follow a logical path through the problem, starting with distribution, combining like terms, and finally carrying out the necessary arithmetic to reach a solution.