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An algebraic expression is a mathematical phrase that contains numbers, variables (letters that represent unknown values), and arithmetic operations such as addition, subtraction, multiplication, and division.

For example, in the expression \(9 a b - 6 a b + 5 a b - 4 a b - 2 a b\), 'ab' is the variable part and the numbers 9, -6, 5, -4, -2 are the coefficients. Each term, like \(9 a b\) or \( -2 a b\), combines a coefficient with a variable part. Algebraic expressions can represent complex relationships and are used to describe patterns, construct functions, and solve equations.

Understanding how to work with these expressions is crucial for solving algebraic problems. Identifying like terms as those sharing the same variable parts allows us to combine them and simplify the entire expression.

The process of simplifying expressions involves reducing an expression to its most basic form without changing its value. In the exercise given, simplification means combining like terms which have the same variable parts.

You can think of like terms as pieces of a puzzle that fit together; when combined, they make the picture (or in this case, the expression) much simpler to understand and work with.

• The first step is to identify and group like terms.
• Next, we add or subtract their coefficients.
• Finally, we attach the common variable part back to the simplified coefficient.

The goal is to consolidate the information given into as straightforward an expression as possible, while retaining its original meaning and value.

At the heart of combining like terms is the use of arithmetic operations: addition, subtraction, multiplication, and division.

In this context, addition and subtraction are used to combine the coefficients of like terms. When we see an expression like \(9 - 6 + 5 - 4 - 2\), we're looking at a sequence of numbers ready to be added or subtracted.

Key Points for Arithmetic Operations:

• Respect the order of operations; but since all operations are at the same level (all addition and subtraction), simply proceed from left to right.
• Keep track of positive and negative signs as they will affect the outcome of the operation.
• Once combined, the number you're left with will be the new coefficient for the variable part that was factored out.

Proper execution of arithmetic operations is essential for accurately simplifying expressions and arriving at the correct solution in algebra.