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In algebra, the distributive property is a fundamental concept. It allows you to multiply a single term across a sum or difference within parenthesis. This property is key when you're dealing with expressions that have brackets, like in the equation we started with:
\[ z-3(z+7)=6(2z+1) \]
For example, to utilize the distributive property on the left side of our equation, we need to distribute the \(-3\) across both terms inside the brackets (\(z\) and \(+7\)). This results in the expression \(-3 \times z - 3 \times 7\). The right side requires you to distribute the 6 similarly.

• The left-hand side becomes \(z - 3z - 21\)
• The right-hand side becomes \(12z + 6\)

Using the distributive property simplifies equations and removes parentheses, making it easier to perform further operations. This step is essential before addressing like terms or variables. Remember, when you see parentheses in an equation, distribute first!

Combining like terms is the process of simplifying mathematical expressions by adding or subtracting terms that have exactly the same variables raised to the same powers.
In our equation, after using the distributive property, we have an expression like this on the left side: \[ z - 3z - 21 = 12z + 6 \]
Only terms that have the same variable part can be combined. So in this example, \(z\) and \(-3z\) are like terms because they both contain the variable \(z\). When you combine them, you get \(-2z\).

• The equation simplifies to \(-2z - 21 = 12z + 6\)

Combining like terms helps to bring similar components together, making the equation less complicated. This step is critical before isolating the variable you wish to solve for.

To solve an algebraic equation, one of the key steps is to isolate the variable. This means getting the variable by itself on one side of the equation, which helps you find its value.
After combining like terms, our goal is to isolate \(z\) in this equation:\[ -2z - 21 = 12z + 6 \]
First, we move all variable terms to one side by adding \(2z\) to both sides:

• This gives us: \(-21 = 14z + 6\)

Next, to simplify further, move the constants to the opposite side. We achieve this by subtracting \(-6\) from both sides:

• We then have: \(-27 = 14z\)

Finally, to solve for \(z\), divide both sides by \(14\):

• The result is \(z = \frac{-27}{14}\)

Isolating the variable step-by-step ensures clarity and accuracy, making sure you properly solve for the unknown component in any algebraic problem.