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Solving equations is an essential skill in mathematics that helps us find unknown values, which in this case is represented by the variable \( z \). When you see an equation like \(-2z = -26\), the goal is to determine the value of \( z \) that makes this equation true.

This involves various techniques such as reversing operations. Since the equation contains multiplication of \(-2\) and \( z \), we will perform the opposite operation, which is division, to both sides of the equation. Finding the right operation to "undo" the operations shown in the equation is key to reaching a solution. Every time you apply an operation to one side, you must also apply it to the other side of the equation, maintaining balance, just like a balanced scale.

The isolation technique is a method used to "solve for" or "find" the unknown variable by getting it alone on one side of the equation. When addressing the initial equation \(-2z = -26\), our objective is to have \( z \) by itself on one side.

To achieve this, use the technique of isolating the variable in several steps:

• Identify the operation currently being applied to the variable \( z \). Here, it is multiplication by \(-2\).
• "Undo" this operation by performing the inverse operation, which involves dividing both sides of the equation by \(-2\).

After performing these steps, you obtain \( z = \frac{-26}{-2} \). So, isolating \( z \) involves systematically and carefully reversing arithmetic operations to achieve a "free" variable.

Variable manipulation is the technical and logical process of rearranging an equation to simplify it or to reveal the value of a variable. In our equation, \(-2z = -26\), after using the isolation technique, we conduct further manipulation to solve for \( z \).

Here's how we proceed:

• We successfully isolated \( z \) and have \( z = \frac{-26}{-2} \).
• Now, perform the calculation: \(-26\) divided by \(-2\) simplifies to \( 13 \), giving the final equation \( z = 13 \).

This manipulation shows how to simplify results effectively and highlights the importance of each mathematical operation—addition, subtraction, multiplication, and division performed properly, ensuring a clear path to understanding and solution.