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When we talk about simplifying expressions in algebra, what we really mean is making the expression as concise as possible, while still keeping its value unchanged. Consider this like tidying up a room - everything is organized without losing anything important.

The process of simplification involves a few key steps:

• Identifying any like terms in the expression that we can combine. Like terms are terms that have the exact same variable (or no variable at all, in the case of constants) and the same power.
• Addition or subtraction of like terms to reduce the number of terms in the expression.

In the given exercise \((2x + 1) + (5x + 8)\), the simplification involves identifying the like terms and combining them to make the expression more compact. The final step is writing the simplified version of the expression, which here is \(7x + 9\).

Simplifying expressions not only makes algebra problems easier to work with, it also lays the foundation for more complex calculations later on.

Like terms are a crucial part of algebraic simplification. To fully grasp this concept, it's important to understand what makes terms "like." These are terms that have:

• The same variable, such as \(x\).
• The same exponent on that variable.

For instance, in the expression \(2x\) and \(5x\) are like terms because both involve the variable \(x\) raised to the first power. Similarly, constants like \(1\) and \(8\) are also considered like terms since they don’t have any variables attached to them.

Recognizing like terms helps in combining them efficiently, which is a crucial step in the simplification process. It eliminates any extra terms, making calculations simpler and algebraic expressions cleaner.

Once you've identified like terms in an algebraic expression, the next step is to combine them. Combining like terms involves straightforward arithmetic operations such as addition or subtraction.
For example, if you have terms like \(2x\) and \(5x\), you combine them as follows:

• Add the coefficients (numbers in front of the variables), so you get \(2x + 5x = 7x\).

Similarly, when combining constants, such as \(1\) and \(8\):

• Simply add them: \(1 + 8 = 9\).

With these calculations, you arrive at the simplified expression: \(7x + 9\). Combining like terms is essentially a cleanup step that simplifies expressions and makes them more manageable for further mathematical operations.