91Ó°ÊÓ

When solving linear equations, one of the primary steps is isolating the variable. In this exercise, the variable is \( z \). Isolating the variable means getting \( z \) alone on one side of the equation.

To do this, you need to move all other terms to the opposite side. In our problem, we began with the equation \( 6 + z = -18 \). The term \( 6 \) is on the same side as \( z \) and must be removed.

We achieve this by performing the inverse operation of addition, which is subtraction. We subtract \( 6 \) from both sides. This step ensures that we maintain the equality of the equation.

So, after subtracting \( 6 \) from both sides, our equation looks like this:

\( 6 + z - 6 = -18 - 6 \).

Now, \( z \) is isolated, allowing us to move to the next step.

After isolating the variable, the next step is to simplify the equation. This means performing basic arithmetic operations to create a more manageable expression.

Let's see how this works in our example of \( 6 + z - 6 = -18 - 6 \). We simplify both sides of the equation by calculating the results of the operations.

On the left side, \( 6 - 6 \) simplifies to 0, so we have \( z \). On the right side, \( -18 - 6 \) simplifies to \( -24 \). The simplified equation becomes:

\( z = -24 \).

Simplifying equations allows us to see the solution in a clear, straightforward manner.

It is important to carefully perform each operation to ensure that the equality remains balanced and the solution is correct.

Inverse operations are mathematical processes that reverse the effects of each other. Understanding these operations is crucial when solving equations because they help us isolate variables.

The main pairs of inverse operations include:

• Addition and subtraction
• Multiplication and division

In our specific problem, the operation we focused on was addition. We had the equation \( 6 + z = -18 \). Addition's inverse operation is subtraction.

By subtracting \( 6 \) from both sides, we effectively removed the 6 from the left-hand side, simplifying the equation to isolate \( z \).

This looks like: \( 6 + z - 6 = -18 - 6 \), which simplifies to \( z = -24 \).

Using inverse operations keeps the equation balanced and helps us logically solve for unknown variables.

Once you become familiar with these operations, solving linear equations will become much easier and more intuitive.