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Understanding the distributive property is key to simplifying algebraic expressions. It's a tool that allows us to multiply a single term by each term within a set of parentheses.
For example, consider the expression \( a(b + c) \). Here, according to the distributive property, we'll multiply \(a\) by \(b\) and then by \(c\) to get \(ab + ac\). This property is particularly useful when dealing with subtraction inside parentheses because we have to distribute negative signs as well.
In the original exercise, \( -6(y + z) \) is a classic distributive property scenario where the \( -6 \) multiplies both \(y\) and \(z\), to get \( -6y - 6z \). Remember, a negative sign in front of the parentheses is distributed as \( -1 \) times each term inside the parentheses, hence the \(y\) becomes \( -6y\) and the \(z\) becomes \( -6z\).

When it comes to simplifying expressions, combining like terms is a basic yet powerful technique. Terms that are ‘like’ have the same variable raised to the same power. Only the coefficients of these terms are different.
For example, \(3x\) and \(5x\) are like terms because they both contain the variable \(x\). Thus, we can add or subtract them to get \(3x + 5x = 8x\). This principle applies regardless of whether the coefficients are positive or negative.
Looking at the original exercise, after distributing, we gathered terms with \(y\) to simplify the expression. The terms \(3y\), \( -6y\), and \(y\) are combined by adding their coefficients. It's like saying you have 3 ‘y’ books, you lose 6 of them, then get 1 more, leaving you with a total of \( -2y\) books. Note that the \(z\) term remains unchanged because there isn't another term like it to combine with.

Algebraic operations form the foundational skills for working through algebra problems. These operations include addition, subtraction, multiplication, and division, applied to both numbers and variables.
When simplifying algebraic expressions, we must follow the conventional order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). In each step, we apply appropriate operations in this sequence to ensure accuracy.
In the context of the given exercise, after the distributive property was applied, we performed algebraic addition and subtraction to combine like terms. A crucial point to note is that unlike numerical operations, we cannot simplify across different variables—each variable or variable combination is like its own individual unit, requiring identical partners for any 'combining' to occur.