91Ó°ÊÓ

An equivalent expression is a new way of writing a mathematical expression that does not change its value. This means that even though the form might look different, the outcome remains the same. For example, if we start with the expression \(-9(ab)\), we can use mathematical rules to rewrite it without changing its value. The result, \(-9(ba)\), looks different but is mathematically equivalent.
The significance of equivalent expressions lies in simplifying problems and making calculations easier. When solving algebraic problems, finding equivalent expressions can be helpful. It often involves applying mathematical properties, like the commutative law, which allows us to alter the order of the variables.

The commutative law is one of the key properties of multiplication. This law states that the order in which you multiply two numbers or variables does not affect the product. Mathematically, you can represent it as follows:

\[ a \times b = b \times a \]
In plain English, this means if you have two quantities being multiplied, swapping their places does not change the result.
For example:
\ ab = ba \
In the context of the exercise, this principle allows us to rewrite the expression \(-9(ab)\) to \(-9(ba)\).
Beyond the commutative law, multiplication also has other interesting properties:

• Associative Property: The way in which numbers are grouped in multiplication does not change the product. For example, \((a \times b) \times c = a \times (b \times c)\).
• Distributive Property: Multiplication distributed over addition. For example, \ a(b + c) = ab + ac \.

All these properties are essential tools for simplifying expressions and solving algebraic equations.

Variable manipulation involves the rearrangement and transformation of algebraic expressions containing variables. It is a fundamental skill in algebra that enables solving for unknowns and simplifying expressions.
In the exercise, we started with \(-9(ab)\) and manipulated the variables inside the parentheses using the commutative law to get \(-9(ba)\).
Here are some practical tips for variable manipulation:

• Always perform operations step by step. This helps to avoid mistakes.
• Use properties like the commutative, associative, and distributive laws to rearrange and simplify expressions.
• Keep track of negative signs, as they can affect the result.
• Practice recognizing patterns. This makes it easier to see how expressions can be simplified.

By mastering variable manipulation, you can gain better control over algebraic expressions and solve complex problems more efficiently.