91Ó°ÊÓ

The distributive property is a fundamental rule used in algebra to help simplify expressions and equations. You apply it by multiplying a single term by each term inside a set of parentheses. This can transform equations, making them easier to solve.
For the equation given, \(4(z-8) = 0\), we need to distribute the 4 across the terms inside the parentheses:

• Multiply 4 by \(z\)
• Multiply 4 by -8

Doing this, we get \(4z - 32 = 0\). This new form of the equation is easier to work with. Remember, distribution allows us to remove the parentheses by spreading the multiplier over each component inside. This step is essential in simplifying algebraic expressions and making them more straightforward to solve.

In algebra, isolating the variable means rearranging the equation so that the variable stands alone on one side of the equation. The goal is to find the value that the variable represents. To do this, perform opposite operations to both sides of the equation step-by-step.

For instance, after applying the distributive property, we have \(4z - 32 = 0\). To isolate \(z\), we first add 32 to both sides of the equation. This gets rid of the constant term on the left, resulting in \(4z = 32\).
Then, to completely isolate \(z\), divide each side by 4, which gives us \(z = 8\). It's important to apply these operations equally to both sides to maintain the equality. By executing these inverse operations, we systematically zero in on the value of the variable.

Once you've found a solution, it's crucial to ensure that it is correct by checking your work. This helps confirm you've followed the right steps and your solution satisfies the original equation.

To check the solution of our example problem, substitute \(z = 8\) back into the original equation, which was \(4(z-8)=0\). Replace \(z\) with 8, giving:

• \(4(8-8)=0\)
• This simplifies down to \(4\times0=0\)

Since both sides of the equation equal zero, our solution \(z = 8\) is verified and correct. Always take this extra step of checking to avoid potential mistakes, especially in more complex equations.