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Equivalent expressions in algebra are expressions that are the same, even if they may look different at first glance. These expressions will give the same result for any values substituted into the variables. For example, you can have two expressions such as \(3 + 2\) and \(5\). Even though they appear different, they both equal 5, thus they are equivalent.

Checking equivalence can involve simplifying both expressions and seeing if they match. In our exercise, this step was evident when simplifying the expressions to see if they share the same form. Equivalent expressions are handy because they allow us to verify our work and confirm that transformations or simplifications maintain the original value or equation.

Simplifying expressions involves combining and reducing expressions to their cleanest form. This might include applying arithmetic operations, getting rid of parentheses through the distributive property, and combining like terms.

For instance, in the original solutions, expressions like \(3 - 3(x + 4)\) are simplified using distribution and combination of terms, resulting in \(-3x - 9\). Simplifying expressions is crucial as it often makes calculations easier and helps in comparing expressions. Start simplifying by handling parentheses, using the distributive property, and then reduce by joining like terms. This process makes the expression more manageable and easier to understand.

The distributive property is an essential algebra concept that allows you to remove parentheses when you see an expression like \(a(b + c)\). It implies that you multiply \(a\) both by \(b\) and \(c\), resulting in \(ab + ac\).

In the exercise solutions, the distributed property was utilized to break down expressions such as \(5 + 2(x - 2)\) into \(5 + 2x - 4\). This step is pivotal because it helps to transition from a longer form with parentheses to a more straightforward, cleaner expression you can work with easily. It showcases how multiplication distributes over addition or subtraction,
making it a powerful tool for simplifying expressions in algebra.

In algebraic expressions, like terms are the terms that have identical variable parts but might differ in their coefficients. They can be easily combined by adding or subtracting their coefficients. The rule is simple: constants with constants, \(x\) terms with \(x\) terms, and so on.

For instance, in expressions like \(-3x - 12 + 3\), we identify like terms such as \(-12\) and \(3\), which are both constants. These terms are combined to give \(-9\), making the expression \(-3x - 9\). Recognizing like terms is critical because combining them is one of the primary tasks in simplifying expressions. When you reduce expressions in this way, you create simpler forms that are easier to understand and equate with other expressions.