Distributive Property Explained Katrina's Expression Simplification
Hey guys! Today, we're diving into a super common algebra problem. Imagine our friend Katrina is tackling this expression: -3a - 4b - 2(5a - 7b). She's simplifying it, and she gets to this step: -3a - 4b - 10a + 14b. The big question is: What allows her to make this jump? Let's break it down and see which property of mathematics is at play here. This article is all about understanding the properties that allow us to manipulate algebraic expressions, making them simpler and easier to work with.
Understanding the Expression
Before we get to the property, let's look closely at what Katrina did. She started with -3a - 4b - 2(5a - 7b) and ended up with -3a - 4b - 10a + 14b. Notice anything? The only real change is that the -2 outside the parentheses has somehow gotten inside and multiplied both terms. That's our clue!
To truly grasp this, let's dissect the expression piece by piece. We have terms involving 'a' and terms involving 'b', and then we have this intriguing part: -2(5a - 7b). This is where the action happens. The -2 is like a gatekeeper, and to let it inside the parentheses, we need a special rule – a mathematical property that gives us permission.
It's essential to understand that in algebra, we're not just moving symbols around randomly. There's a logic, a set of rules that we must follow. These rules, or properties, ensure that our manipulations are valid and that the final result is equivalent to the original expression. In this case, Katrina's move is a classic example of applying one of these fundamental properties. So, which one is it?
The Distributive Property: The Key Player
The property that Katrina used is the distributive property. This property is a fundamental concept in algebra, and it's something you'll use all the time. In simple terms, the distributive property allows you to multiply a single term by multiple terms inside a set of parentheses. It's like sharing the love (or, in this case, the multiplication) with everyone inside the group.
The distributive property can be stated like this: a(b + c) = ab + ac. See how the 'a' gets distributed to both 'b' and 'c'? That's exactly what Katrina did. She distributed the -2 to both the 5a and the -7b inside the parentheses.
Let's see it in action with our expression. Katrina had -2(5a - 7b). Using the distributive property, she multiplied -2 by 5a to get -10a, and she multiplied -2 by -7b to get +14b. Remember that a negative times a negative is a positive! This is a crucial detail to keep in mind when applying the distributive property, especially when dealing with negative signs. The distributive property is not just a rule; it's a tool that empowers us to rewrite expressions in a way that makes them easier to simplify and solve.
Why Not the Other Properties?
Okay, so we've identified the distributive property, but why not the other options? Let's take a quick look at why the associative and commutative properties don't fit in this scenario. It's important to understand not only what the correct answer is but also why the other options are incorrect.
Associative Property
The associative property deals with grouping. It says that when you're adding or multiplying, you can group the numbers in any way you want without changing the result. For example, (a + b) + c = a + (b + c). Notice that the order of the terms (a, b, and c) stays the same; only the parentheses move. In Katrina's problem, we're not just regrouping; we're actually performing a multiplication operation across the parentheses. The associative property doesn't cover that kind of situation.
Commutative Property
The commutative property is all about order. It says that you can change the order of terms when adding or multiplying without affecting the outcome. For example, a + b = b + a and a * b = b * a. Again, in Katrina's simplification, we're not just rearranging terms; we're distributing a factor across a sum. The commutative property simply doesn't address the operation Katrina performed.
Understanding why these properties don't apply is just as important as knowing why the distributive property does. It reinforces your understanding of each property's specific role in mathematical operations. In this case, the distributive property is the star of the show because it directly addresses the multiplication of a term across a set of parentheses.
Final Answer: C. the Distributive Property
So, the answer is C. the distributive property. Katrina used the distributive property to multiply the -2 by both terms inside the parentheses, which is the key step in simplifying the expression.
This problem is a fantastic example of how important it is to understand the basic properties of math. They're the foundation upon which more complex concepts are built. If you have a solid grasp of these properties, simplifying expressions becomes much easier and more intuitive.
Remember, guys, math isn't just about memorizing formulas; it's about understanding the rules and knowing when and how to apply them. In this case, Katrina's application of the distributive property shows us a perfect example of how these rules work in action. Keep practicing, and you'll become simplification pros in no time!
Let's delve deeper into the realm of mathematical properties, especially in the context of simplification. Katrina's problem beautifully illustrates how a single property can be the key to unlocking a simplified expression. But what does it really mean to
