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The distributive property is a key algebraic principle. It states that multiplying a single term by terms inside a parenthesis distributes the multiplication across each term within the parenthesis. In mathematical terms, for any numbers a, b, and c, we have: \( a(b + c) = ab + ac \).

Consider the expression: \(-(12z + 1) + 2(7z - 5)\). Here, we need to distribute the negative sign and the 2.

First, distribute the -1 across \(12z + 1\), which gives \(-1(12z) + (-1)(1)\), resulting in \(-12z - 1\).

Next, distribute the 2 across \(7z - 5\), which is \(2(7z) + 2(-5)\), resulting in \(14z - 10\). Combining these, we get \(-12z - 1 + 14z - 10\).

Utilizing the distributive property ensures we handle each term correctly, preventing errors.

Combining like terms simplifies expressions by adding or subtracting coefficients of similar variables. 'Like terms' are terms with the same variable raised to the same power.

Consider our simplified expression: \(-12z - 1 + 14z - 10\).

Identify the like terms:

• Terms with z: \(-12z\) and \(14z\)
• Constant terms: \(-1\) and \(-10\)

To combine the like terms, add or subtract the coefficients:
\(-12z + 14z = 2z\),
and add the constants: \(-1 - 10 = -11\).

This gives us the simplified result: \(2z - 11\).

Combining like terms reduces the complexity of algebraic expressions and helps reach a final simplified form.

Distributing a negative sign is vital in correctly simplifying expressions. It means multiplying every term inside the parentheses by -1.

Let's look again at the expression \(-(12z + 1)\):

• This becomes \((-1) \cdot 12z + (-1) \cdot 1\), resulting in \(-12z - 1\).

Distributing the negative sign properly ensures each term is correctly modified, making further steps easier.

Failure to distribute the negative sign correctly can lead to errors. Always remember to multiply each term within the parentheses by -1 when you see a negative sign outside.

Thus, handling the negative sign correctly at the start helps pave the way for accurate algebraic simplification.