Solving Equations Verona's Math Problem Explained
Hey everyone! Let's dive into a fun math problem today. We're going to help Verona solve an equation. It's like being a math detective, and we're going to crack the code together. So, grab your thinking caps, and let's get started!
Understanding the Equation: -3 + 4x = 9
At the heart of our puzzle is the equation -3 + 4x = 9. Now, what does this even mean? Think of it like a balanced scale. On one side, we have '-3 + 4x', and on the other side, we have '9'. The equals sign (=) tells us that both sides are perfectly balanced. Our mission, should we choose to accept it, is to figure out what 'x' needs to be to keep this balance. Keywords here are equation, balanced scale, and solving for x.
The Goal: Isolating the Variable Term
Verona's mission, and ours too, is to isolate the variable term. What's a variable term, you ask? It's simply the part of the equation that has our mystery 'x' in it. In this case, it's '4x'. To isolate it means to get it all by itself on one side of the equation. Imagine separating the '4x' from the '-3' so it can shine on its own. This is a crucial step in solving any algebraic equation, guys. You'll see this concept pop up again and again, so mastering it now is like leveling up your math skills!
The Addition Property of Equality: Our Secret Weapon
Here's where our superpower comes in: the addition property of equality. This fancy term simply means that we can add the same number to both sides of an equation without messing up the balance. It's like adding the same weight to both sides of our scale – it stays perfectly level. This property is a cornerstone of algebra and is your best friend when you're trying to solve for a variable. Remember, we are keeping the equation balanced so that we don’t change the solution. This principle is fundamental and allows us to manipulate equations strategically.
Cracking the Code: What Number to Add?
Now for the million-dollar question: which number should Verona add to both sides? Remember, we want to isolate '4x'. Right now, we have '-3 + 4x'. What if we could somehow get rid of that '-3'? Think of it like having a pesky pebble in your shoe – you want to get rid of it! Well, we can! We can use the addition property of equality to do it. The opposite of '-3' is '+3'. So, if we add '+3' to both sides, the '-3' on the left side will disappear. Magic, right? Well, it's math magic!
Let’s break it down step by step:
- Start with the equation: -3 + 4x = 9
- Add 3 to both sides: -3 + 4x + 3 = 9 + 3
- Simplify: 4x = 12
See what happened? The '-3' and '+3' on the left side canceled each other out, leaving us with '4x' all by itself. That's exactly what we wanted! This step is crucial for isolating the variable and moving closer to the solution.
Why the Other Options Don't Work
Let's quickly look at why the other options aren't the right choice:
- A. -4: Adding -4 wouldn't cancel out the -3. It would actually make it even more negative! (-3 + -4 = -7). So, adding -4 wouldn’t get us closer to isolating '4x'.
- C. -3: Adding -3 would make the left side even more negative (-3 + -3 = -6). This would move us further away from our goal of isolating '4x'.
- D. 4: Adding 4 to both sides would give us -3 + 4 + 4x = 9 + 4, which simplifies to 1 + 4x = 13. While this is a valid mathematical step, it doesn't help us isolate the '4x' term directly. We still have that '+1' hanging around.
The Answer: B. 3
The correct answer is B. 3. Adding 3 to both sides of the equation is the key to unlocking this math puzzle. It cancels out the -3, leaving us with the isolated variable term '4x'. High five, guys! We're on our way to solving for 'x'!
Taking it Further: Solving for x
Now that we've isolated '4x', let's finish the job and solve for 'x' completely! Remember our equation after adding 3 to both sides? It's 4x = 12. Now we need to get 'x' all by itself. Right now, it's being multiplied by 4. So, what's the opposite of multiplying by 4? Dividing by 4, of course! We use the division property of equality here which states that you can divide both sides of an equation by the same non-zero number without changing the solution.
The Division Property of Equality: Another Tool in Our Arsenal
The division property of equality is like the addition property's sibling. It says that we can divide both sides of an equation by the same number (as long as it's not zero!) and maintain the balance. This is our ticket to finally revealing the value of 'x'.
Finishing the Puzzle: Dividing Both Sides by 4
Let's divide both sides of our equation (4x = 12) by 4:
- Start with: 4x = 12
- Divide both sides by 4: 4x / 4 = 12 / 4
- Simplify: x = 3
Eureka! We found it! The value of 'x' that makes the equation true is 3. We've cracked the code, solved the puzzle, and conquered the equation! Give yourselves a pat on the back, guys. You've earned it!
Checking Our Work: The Final Step
But wait, we're not done yet! A good math detective always checks their work. It's like making sure you've dotted your i's and crossed your t's. To check our answer, we simply plug 'x = 3' back into the original equation and see if it holds true.
Original equation: -3 + 4x = 9
Substitute x = 3: -3 + 4(3) = 9
Simplify: -3 + 12 = 9
Simplify further: 9 = 9
It works! The equation balances perfectly when x = 3. We've not only solved the equation, but we've also confirmed our solution. Double score!
Wrapping Up: Math Detective Skills Acquired!
So, there you have it! We helped Verona solve the equation -3 + 4x = 9 by understanding the importance of isolating the variable term and using the addition and division properties of equality. We learned that adding 3 to both sides was the key to getting started, and we even solved for 'x' and checked our answer. You've leveled up your math skills today, guys. Keep practicing, keep exploring, and keep cracking those math codes! You're all math detectives now!
This journey through Verona's equation highlights the systematic approach to solving algebraic problems. By identifying the key operations needed to isolate the variable and employing the properties of equality, we transformed a seemingly complex problem into a straightforward solution. Remember, math isn't just about finding the answer; it's about understanding the process and the underlying principles. This understanding empowers you to tackle even more challenging equations and problems in the future. So, keep your thinking caps on and continue exploring the fascinating world of mathematics!
Remember, every equation is a puzzle waiting to be solved, and with the right tools and techniques, you can become a master puzzle-solver. Keep practicing, keep learning, and most importantly, keep having fun with math!
