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When simplifying algebraic expressions, one fundamental technique is the use of the distributive property. It allows you to multiply a single term by a group of terms within parentheses. The standard form of the distributive property is expressed as \(a(b + c) = ab + ac\), where \(a\), \(b\), and \(c\) represent any algebraic terms.

For example, if you have the expression \(2(x + 3)\), you would apply the distributive property by multiplying \(2\) by both \(x\) and \(3\), which results in the expression \(2x + 6\). However, the distributive property also applies when we're dealing with negative numbers or subtraction. This is particularly handy when simplifying expressions that involve subtracting a group of terms, such as in the given exercise \( -(3x - 4y + z) \).

To simplify an expression means to make it as straightforward and compact as possible. Combining like terms is a critical step in this process. Like terms are terms that have the same variables raised to the same power. For instance, \(2x\) and \(5x\) are like terms because they both have the variable \(x\) to the first power.

When simplifying, you add or subtract the coefficients (the numbers in front of the variables) of like terms. So \(2x + 5x\) simplifies to \(7x\). Simplification may also involve getting rid of parentheses using the distributive property, as shown in the initial exercise. After distributing, it becomes easier to identify and combine like terms to reach the simplest form of the expression.

Understanding how to distribute a negative sign across an expression is essential in algebra. When a negative sign precedes parentheses, it affects every term inside, and you can think of it as \( -1 \times (expression)\).

If we consider our example \( -(3x - 4y + z) \), distributing the negative sign is the same as multiplying each term by -1. Thus, \(3x\) becomes \( -3x\), \( -4y\) becomes \( +4y\), and \(z\) becomes \( -z\). Remember, when a negative sign is distributed to a negative term, it results in that term becoming positive, as two negatives make a positive. This concept is vital when simplifying expressions that start with a negative sign or contain subtraction.