91Ó°ÊÓ

In order to solve any linear equation, the first step is to isolate the variable. This means getting the variable you are solving for, like \( r \), alone on one side of the equation. This allows us to directly find its value.

Here is how it's done in the exercise, given the equation:

-9r = -54

To isolate \( r \), we need to get rid of the coefficient of \( r \), which is -9.
We achieve this by performing the opposite operation. Since \( r \) is being multiplied by -9, we will divide both sides of the equation by -9. This simplifies our equation and helps us in solving it.

The operation looks like this:

\[ \frac{-9r}{-9} = \frac{-54}{-9} \]

This keeps the equation balanced and isolates the variable \( r \) on the left side.

Now we can move on to the next step, which is handling the division.

When both sides of the equation are divided by the same number, it keeps the balance of the equation. Division helps us simplify the problem further.

As applied to our example, after dividing both sides of the equation by -9, we get:

\[ r = \frac{-54}{-9} \]

Next, we need to perform the division operation. Dividing numbers can sometimes be tricky, especially when they involve negative signs.

• A negative divided by a negative results in a positive.
• Make sure to always include the negative signs in your calculations.

In our case, -54 divided by -9 is:

\( \frac{-54}{-9} = 6 \)

After performing this step, the division part of solving the equation is complete.
Now let's move on to simplifying the result.

After performing the necessary operations, we get a value for our variable. The final step is to simplify this value if it can be simplified.

For the equation, we have reached:

\( r = 6 \)

Since 6 is already in its simplest form, there is no need for further simplification.

Simplifying means making sure that the answer is as short and comprehensible as possible. If the result was a fraction, we would reduce it to its simplest terms. This ensures clarity and correctness of the solution.

• Always double-check if the final value can be simplified further.

Now we have our final answer for the variable \( r \) in the given equation: \( r = 6 \).
It’s important to follow these steps in solving linear equations. It ensures accuracy and helps understand the process clearly.