Distributive Property: Which Equation Shows It?

Hey guys! Let's break down the distributive property and figure out which of these equations shows it in action. It's a fundamental concept in mathematics, and understanding it will seriously level up your algebra game. We'll go through each option, explain why it is or isn't distributive, and make sure you're crystal clear on what's going on.

Understanding the Distributive Property

The distributive property is all about how multiplication interacts with addition or subtraction. Basically, it lets you multiply a single term by two or more terms inside a set of parentheses. The general form looks like this: a(b + c) = ab + ac. What's happening here is that the term outside the parentheses (a) gets distributed to each term inside the parentheses (b and c). This makes calculations easier, especially when you're dealing with variables. For example, if you have 2*(x + 3), you can distribute the 2 to get 2x + 23, which simplifies to 2x + 6. This property is super useful in algebra for simplifying expressions and solving equations. Understanding the distributive property is also crucial for more advanced math topics, such as factoring polynomials and working with complex numbers. So, grasping this concept early on will save you a lot of headaches down the road. It's like having a secret weapon in your math arsenal that you can use to tackle all sorts of problems. The distributive property also works with subtraction: a(b - c) = ab - ac. The key is that you're multiplying the term outside the parentheses by each term inside, making sure to keep the correct sign. This is super handy when you're dealing with expressions like 3*(y - 2), which simplifies to 3y - 6. Remember, the distributive property is your friend when it comes to simplifying and solving equations. So, keep practicing, and you'll become a master of distribution in no time!

Now, let's examine each option to see which one showcases this property:

1. $2+3=3+2$

2. $2 \times 3=3 \times 2$

3. $2+0=2$

4. $2 \times(3+4)=2 \times 3+2 \times 4$

Detailed Analysis of Each Option

Let's dive into each option and see why it either demonstrates the distributive property or why it showcases something else entirely. We'll break down the math and the logic so you can clearly see what's happening in each case.

Option 1: $2 + 3 = 3 + 2$

This equation, $2 + 3 = 3 + 2$, illustrates the commutative property of addition. The commutative property states that you can change the order of addends without changing the sum. In simpler terms, it doesn't matter if you add 2 and 3 or 3 and 2; the result will always be the same (which is 5). This property is fundamental in arithmetic and algebra, allowing you to rearrange terms in an addition problem to make it easier to solve or simplify. For instance, if you have an expression like x + 5 + 2, you can rewrite it as x + 2 + 5 to group like terms. Understanding the commutative property helps streamline calculations and simplifies algebraic manipulations. It's one of the basic building blocks of mathematical operations, ensuring that the order of addition doesn't affect the outcome. It's also super helpful when dealing with complex equations where rearranging terms can make the problem more manageable. So, while it's a useful property to know, it's not the distributive property we're looking for.

Option 2: $2 \times 3 = 3 \times 2$

Similarly, the equation $2 \times 3 = 3 \times 2$ demonstrates the commutative property of multiplication. Just like with addition, the commutative property of multiplication means that the order in which you multiply numbers doesn't change the product. Whether you multiply 2 by 3 or 3 by 2, the result is always 6. This property is incredibly useful in various mathematical contexts, especially when simplifying expressions or solving equations. For example, in algebra, if you have 2 * x * 5, you can rewrite it as 5 * x * 2 to rearrange the terms and simplify the expression. Knowing this property can help you solve problems more efficiently and accurately. It's another foundational concept that makes mathematical operations more flexible and easier to handle. So, while it's an important property to recognize, it's not the distributive property we're trying to identify.

Option 3: $2 + 0 = 2$

The equation $2 + 0 = 2$ illustrates the identity property of addition. This property states that any number plus zero equals that number. Zero is the additive identity because adding it to any number doesn't change the number's value. This property is a basic concept in arithmetic and is essential for understanding more complex mathematical operations. For example, when solving equations, adding zero to one side can sometimes help simplify the equation without changing its balance. The identity property of addition is a fundamental principle that underlies many algebraic manipulations. It's a simple yet powerful concept that ensures that adding nothing to a number leaves the number unchanged. This property is often used implicitly in mathematical calculations and is a key component of understanding number systems. So, while this is another key property, it's not the distributive property.

Option 4: $2 \times (3 + 4) = 2 \times 3 + 2 \times 4$

This equation, $2 \times (3 + 4) = 2 \times 3 + 2 \times 4$, perfectly demonstrates the distributive property. Here's why: On the left side, you have 2 multiplied by the sum of 3 and 4. On the right side, you have 2 multiplied by 3, plus 2 multiplied by 4. Let's break it down further:

• Left side: $2 \times (3 + 4) = 2 \times 7 = 14$
• Right side: $2 \times 3 + 2 \times 4 = 6 + 8 = 14$

Both sides are equal, showing that the 2 has been distributed across the terms inside the parentheses. This is exactly what the distributive property is all about! It allows you to break down multiplication over addition (or subtraction) into separate multiplication operations, making complex calculations easier to manage. The distributive property is a fundamental concept in algebra and is used extensively in simplifying expressions and solving equations. It's also a crucial tool for understanding more advanced mathematical concepts. So, this is our winner!

Final Answer

Therefore, the correct answer is:

4. $2 \times (3+4)=2 \times 3+2 \times 4$