91Ó°ÊÓ

To solve linear equations effectively, you need to isolate the variable. This means getting the variable (in our case, \(n\)) alone on one side of the equation. Let's look at the problem again:

\(8n + 2 = 6n + 12\).

Start by moving all terms involving \(n\) to one side. This is achieved by subtracting \(6n\) from both sides:

\(8n + 2 - 6n = 6n + 12 - 6n\).

This simplifies to \(2n + 2 = 12\).
Now our goal is to get \(n\) by itself. We subtract 2 from both sides:

\(2n + 2 - 2 = 12 - 2\).

Simplifying this we get:

\(2n = 10\).
Finally, divide both sides by 2 to isolate \(n\):
\(n = 10 / 2\)
giving us the final answer, \(n = 5\).

When we talk about combining like terms, we're referring to the process of simplifying expressions by adding or subtracting terms that have the same variable. Let's walk through our example:

Starting point was:
\(8n + 2 = 6n + 12\).

Our earlier goal was to move all \(n\) terms to one side. By subtracting \(6n\), we get:
\(8n - 6n + 2 = 12\).
Here, \(8n - 6n\) are 'like terms' because they both have the same variable \(n\), so we combine them:
\(2n + 2 = 12\).
This makes our equation simpler and easier to work with.

Inverse operations are operations that undo each other, such as addition and subtraction, or multiplication and division. We use these to solve equations by simplifying step-by-step until we isolate the variable. In the example:

First, we had the equation:
\(8n + 2 = 6n + 12\).

We subtracted \(6n\) to move terms with \(n\) to one side which is one of the inverse operations:
\(8n - 6n + 2 = 6n - 6n + 12 = 2n + 2 = 12\).
Next, to get \(2n\) by itself, subtract 2 from both sides:
\(2n + 2 - 2 = 12 - 2\).
This simplifies to:
\(2n = 10\).
Finally, we use the inverse operation of multiplication, which is division, to solve for \(n\):
\(2n / 2 = 10 / 2\).
This gives us the final result:
\(n = 5\).

These steps show how inverse operations help us steadily break down and solve equations.