Equality Property: Solving 3z/2 - 5 = 10

Hey guys! Ever wondered what makes those equation-solving steps so legit in math? It all comes down to the properties of equality. Let's break down a common problem and see which property gives us the green light to do what we do. We will dive deep into understanding the properties of equality. More specifically, we'll dissect an example equation to pinpoint exactly which property is being used. So, grab your thinking caps, and let's get started!

Understanding Properties of Equality

In the world of algebra, the properties of equality are like the rulebook for keeping equations balanced. Think of an equation like a scale: whatever you do to one side, you absolutely have to do to the other to keep it even. These properties are the reasons why we can manipulate equations the way we do. They ensure that every step we take is mathematically sound and that we arrive at the correct solution. Before we jump into our specific problem, it's super helpful to have a quick rundown of some of the most common properties. This way, when we see them in action, we can quickly identify them and understand why they're being used. Let's review a few key players:

• Addition Property of Equality: This one's pretty straightforward. It says that if you add the same number to both sides of an equation, the equation remains balanced. For example, if you have a = b, then a + c = b + c. Simple as that!
• Subtraction Property of Equality: Similar to addition, this property states that you can subtract the same number from both sides of an equation without changing the equality. So, if a = b, then a - c = b - c.
• Multiplication Property of Equality: You guessed it! If you multiply both sides of an equation by the same non-zero number, the equation remains equal. If a = b, then a * c = b * c (as long as c isn't zero).
• Division Property of Equality: This is the flip side of multiplication. Dividing both sides of an equation by the same non-zero number keeps the equation balanced. If a = b, then a / c = b / c (again, c can't be zero).
• Reflexive Property of Equality: This one's a bit more abstract but still important. It simply says that any quantity is equal to itself. So, a = a.
• Symmetric Property of Equality: This property lets you flip the sides of an equation. If a = b, then b = a.
• Transitive Property of Equality: This one's a chain reaction. If a = b and b = c, then a = c.
• Substitution Property of Equality: If two things are equal, you can substitute one for the other in any equation or expression. If a = b, then a can replace b (and vice versa) in any equation.

The Equation at Hand: 3z/2 - 5 = 10

Alright, let's get our hands dirty with the equation we're tackling: 3z/2 - 5 = 10. This is a classic algebraic equation, and our mission is to isolate 'z' and figure out what value makes this equation true. But we're not just interested in the answer; we want to understand why we're allowed to do each step along the way. Think of it like this: we're not just following a recipe; we're learning the chemistry of the ingredients. So, let's break down the steps:

Step 1: 3z/2 - 5 + 5 = 10 + 5

This is the crucial step we need to analyze. What exactly happened here? Well, if you look closely, we added '5' to both sides of the equation. Notice how the original equation, 3z/2 - 5 = 10, has been transformed. We've taken that initial state and performed an operation that maintains the balance. Adding the same value to both sides is a fundamental move in algebra, and it's the key to isolating our variable, 'z'. But which property of equality allows us to do this? This is where our earlier review comes in handy. Remember the Addition Property of Equality? It states that adding the same number to both sides of an equation preserves the equality. And that's precisely what's happening here!

By adding '5' to both sides, we're not disrupting the balance of the equation. We're simply performing the same operation on both sides, ensuring that the left side remains equal to the right side. This is a core principle in solving equations, and it's rooted directly in the Addition Property of Equality. This step is all about strategically eliminating the '-5' on the left side. By adding '+5', we're effectively neutralizing it, bringing us closer to isolating the term with 'z'. This is a common tactic in algebra: using inverse operations to simplify equations. The addition property is not just a theoretical rule; it's a practical tool that helps us move closer to our solution.

Step 2: 3z/2 = 15

Now, let's look at the result of our addition: 3z/2 = 15. This equation is a simplified version of our original one. We've successfully eliminated the constant term on the left side, bringing us one step closer to isolating 'z'. But how did we get here? This step is a direct consequence of the previous one. When we added '5' to both sides of the equation 3z/2 - 5 + 5 = 10 + 5, the '-5' and '+5' on the left side canceled each other out. This is a basic arithmetic operation, but it's crucial for understanding the flow of the solution. On the right side, '10 + 5' simplifies to '15'. So, by performing the addition, we've transformed the equation into a more manageable form. This simplified equation, 3z/2 = 15, is now our new focus. We've made progress, but we're not quite there yet. We still need to isolate 'z', and we'll do that by applying other properties of equality in the subsequent steps.

Identifying the Correct Property

So, the question asks: Which property justifies the step 3z/2 - 5 + 5 = 10 + 5? We've already nailed it down: it's the Addition Property of Equality. This property is the backbone of this step, allowing us to add '5' to both sides without messing up the equation's balance. Now, let's quickly look at why the other options aren't the right fit. This is a great way to reinforce our understanding and make sure we're crystal clear on the properties.

• Division Property of Equality: This property involves dividing both sides of the equation by the same non-zero number. We didn't divide anything in this step, so this one's out.
• Multiplication Property of Equality: This property is all about multiplying both sides by the same number. Again, no multiplication happened here, so it's not the answer.

Why the Addition Property of Equality Matters

Guys, the Addition Property of Equality isn't just some abstract rule we learn in algebra class. It's a fundamental tool that empowers us to solve equations. It allows us to manipulate equations strategically, moving terms around and isolating variables until we arrive at the solution. Think about it: without this property, we'd be stuck! We wouldn't be able to add numbers to both sides, and our equation-solving toolbox would be severely limited. The beauty of this property lies in its simplicity and its power. It's a straightforward concept – add the same thing to both sides – but it unlocks a world of possibilities in algebra. It's a cornerstone of equation solving, and it's essential for anyone who wants to master algebra.

Moreover, the Addition Property of Equality is a building block for more advanced mathematical concepts. It's used extensively in higher-level algebra, calculus, and beyond. Understanding this property deeply will set you up for success in your future math endeavors. So, it's not just about solving this specific equation; it's about building a solid foundation for your mathematical journey.

In summary, the Addition Property of Equality is a game-changer. It allows us to maintain balance while we manipulate equations, and it's a key ingredient in the recipe for solving algebraic problems. So, next time you're adding the same number to both sides of an equation, remember the Addition Property of Equality and give yourself a mental high-five for using it!

Conclusion

So, there you have it! We've successfully identified the property of equality that justifies the step in solving the equation 3z/2 - 5 = 10. It's the Addition Property of Equality, the superhero that allows us to add the same number to both sides without breaking the balance. Understanding these properties is key to conquering algebra, so keep practicing and keep exploring! You've got this!