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Isolating the variable is a critical first step in solving linear equations. It means manipulating the equation such that the variable you are trying to solve for is on one side of the equation, and everything else is on the other side.

For example, in the equation \(5x = 45\), the goal is to get \(x\) by itself on one side of the equal sign. To do this, we look at what is happening to our variable. Here, \(x\) is being multiplied by 5. To undo this multiplication and isolate \(x\), we need to divide both sides of the equation by 5. This step is based on the principle of balance, meaning whatever you do to one side, you must do to the other to maintain equality.

By dividing both sides by 5, we effectively get \(\frac{5x}{5} = \frac{45}{5}\), which simplifies to \(x=9\). Now, the variable is isolated, and the equation is simplified enough to solve.

Once the variable is isolated, the next step is to simplify the equation. Simplification makes the solution much more apparent and manageable. It often involves arithmetic operations such as addition, subtraction, multiplication, division, and canceling out like terms.

In our example, after isolating \(x\), we had the equation \(x = 45/5\). To simplify, we perform the division, which is straightforward: 45 divided by 5 equals 9. Hence, the equation becomes \(x = 9\).

It's essential to do these operations carefully to avoid errors. When simplifying, remember to perform the same operation to all terms involved and watch out for negative signs or fractions, as they are common sources of mistakes.

After solving a linear equation, checking the solution is a vital final step to ensure the answer is correct. This involves substituting the solved value back into the original equation to see if the left-hand side (LHS) equals the right-hand side (RHS).

For the equation we've been discussing, \(5x = 45\), after simplifying, we found \(x = 9\). To check this, plug \(9\) back into the original equation: \(5 * 9 = 45\). This simplifies to \(45 = 45\), which is a true statement, confirming that our solution, \(x = 9\), is indeed correct.

Always make a habit of checking your answers. It not only verifies your solution but also provides a deeper understanding of the problem by reinforcing the concept that what you do to one side of an equation, you must do to the other to maintain balance.