91Ó°ÊÓ

One of the fundamental steps in solving linear equations is to isolate the variable. This means getting the variable on one side of the equation and everything else on the opposite side. In our exercise, we started with the equation \(9x + 5.5 = 10x\). To isolate the variable \(x\), we needed to remove \(9x\) from the left side.

This is accomplished by performing the same operation (in this case, subtraction) on both sides of the equation. By subtracting \(9x\) from both sides, we simplify our equation without changing its overall balance.

Remember that whatever you do to one side of an equation, you must do to the other. This principle is crucial for maintaining equality and achieving accurate results.

After isolating the variable terms, the next essential step is simplifying the equation. This involves performing basic arithmetic operations to combine like terms and achieve a more manageable expression. Continuing from our example, after subtracting \(9x\) from both sides, the equation was reduced to \(5.5 = 10x - 9x\).

On the right side of the equation, \(10x - 9x\) simplifies to \(x\). Thus, the equation becomes \(5.5 = x\). The simplification process is about transforming the equation into its simplest form where solving for the variable becomes straightforward.

Simplifying equations not only clears complexity but also reveals the solution directly, as we see with \(x = 5.5\) in this problem.

The final step is to verify the solution. This is an important part of solving any equation to ensure the solution is correct. We substitute our solution for \(x\) back into the original equation. With our solution \(x = 5.5\), we substitute and check:

- Calculate the left side: \(9(5.5) + 5.5\)
- Calculate the right side: \(10(5.5)\)

Both sides should equal when the solution is correct, which they do here, confirming the solution \(x = 5.5\) is valid.

Verification gives us confidence that our isolated and simplified variable truly satisfies the original equation. This is essential to avoid potential arithmetic mistakes in complex equations.