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When you solve an equation, one of the most important goals is to isolate the variable. This means getting the variable (like \( z \)) by itself on one side of the equation. By doing this, you can easily determine its value. Begin with the given equation: \[ -z - 8 = 21 \] The first step is to remove or cancel out the constant term (in this case, \( -8 \)). You can do this by performing the opposite (inverse) operation. Since \( -8 \) is being subtracted, you add 8 to both sides of the equation to cancel it out: \[ -z - 8 + 8 = 21 + 8 \] This simplifies to: \[ -z = 29 \] Now, the variable term \( -z \) is isolated on one side. This method allows you to focus on solving for the variable itself.

Once you have found a solution to the equation, it's very important to verify, or check, if your solution is correct. This ensures that you haven't made any mistakes in your calculations. Let's check our solution by substituting \( z = -29 \) back into the original equation. The original equation is: \[ -z - 8 = 21 \] Substituting \( -29 \) for \( z \), we get: \[ -(-29) - 8 = 21 \] This simplifies as follows: \[ 29 - 8 = 21 \] And further to: \[ 21 = 21 \] Since both sides of the equation are equal, our solution \( z = -29 \) is confirmed to be correct. Remember, always plug your solution back into the original equation to check its validity.

Understanding inverse operations is crucial for solving equations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. In our problem, we used inverse operations twice:

• First, we added 8 to both sides of the equation to isolate the variable term:
\[ -z - 8 + 8 = 21 + 8 \] This step canceled out the \( -8 \) on the left side.
• Second, we multiplied both sides by \( -1 \) to isolate \( z \):
\[ -1(-z) = -1(29) \] These steps allowed us to successfully find that \( z = -29 \). By understanding and using inverse operations, you can systematically solve for the variable and simplify your equation-solving process. It's like unraveling a math puzzle one step at a time.