91Ó°ÊÓ

The distributive property is a cornerstone of algebra that allows us to simplify expressions. It's the rule that enables us to distribute, or multiply, a single term across the terms within a parenthesis. For instance, in the expression \(3(x+2)\), we apply the distributive property by multiplying 3 with each term inside the parentheses—\(x\) and 2—resulting in \(3x + 6\).

It's like sharing equally among friends! Imagine you have 3 bags of candy and each bag has a piece of chocolate and two gummies (\

After distributing, we often find ourselves with an expression that has similar terms, which we call 'like terms'. Like terms are expressions that have the same variables raised to the same power. In our example, \(3x\) and \(-5x\) are like terms because they both have the variable \(x\) to the first power.

Imagine you're organizing a fruit bowl. You wouldn't mix apples with oranges because they're different. Similarly, when simplifying expressions, you combine apples with apples—in our case, \(x\)'s with \(x\)'s! By combining the like terms \(3x\) and \(-5x\), we get a new term: \(-2x\) (like combining 3 apples with 5 apples taken away, you end up with 2 apples short). With constants (numbers without variables), we combine 6 and 35 to get 41.

• Add \(3x\) and \(-5x\): \(3x - 5x = -2x\)
• Add constants \(6\) and \(35\): \(6 + 35 = 41\)

Algebraic manipulation involves rearranging and simplifying expressions using various techniques. It's like solving a puzzle where the pieces are numbers and algebraic terms. Once we've distributed our terms and combined like terms, we should have a simplified expression that clearly communicates the result without extra baggage. In the provided exercise, after applying the distributive property and combining like terms, we manipulate the expression to its simplest form \(-2x + 41\).

This process is crucial for understanding more complex algebraic concepts and solving equations. It's how we tidy up our math work, so the answer is clear and easy to understand. In a sense, algebraic manipulation is the art of making math expressions neat and elegant!

• Organize terms: \(-2x + 41\) is neater than the original expression
• Simplify: A simpler expression makes it easier to understand or to use in further calculations