Hey guys! Today, we're diving into a super common type of math problem – solving equations. But we're not just going to find the answer; we're going to understand why each step works. Let's break down this equation and see what's really going on. Our main goal here is to clarify the justification for Step 2 in a given solution process. We'll explore the properties of equality that allow us to manipulate equations while keeping them balanced. So, grab your thinking caps, and let's get started!
The Equation at Hand
We're given the equation:
And the first couple of steps in solving it:
Step 1: $7x - 25 = 4x - 1$ Step 2: $7x = 4x + 24$
The big question we're tackling today is: What allows us to go from Step 1 to Step 2? What's the magic behind it? It's not magic, of course; it's math! We need to pinpoint the exact mathematical principle that justifies this transformation. To truly master algebra, it's not enough to just follow the steps; you need to grasp the why behind them. This deeper understanding will empower you to tackle even the trickiest equations with confidence. Remember, understanding the justification is key to building a strong foundation in mathematics. So, let's unravel this mystery together and make sure we're crystal clear on how we can manipulate equations while maintaining their balance and accuracy.
Decoding Step 1: Combining Like Terms
Before we zoom in on Step 2, let's quickly recap what happened in Step 1. This helps us build a solid foundation for understanding the next leap in the solution. In the original equation,
we had a couple of terms with 'x' on the left side: 10x and -3x. Step 1 is a classic move in algebra: combining like terms. This simplifies the equation and makes it easier to work with. Think of it like tidying up your room – putting similar things together makes everything less cluttered.
Combining like terms is based on the distributive property and the commutative property of addition. Basically, we're saying that 10 of something minus 3 of that same thing leaves us with 7 of that thing. In our case, that 'thing' is 'x'. So, 10x - 3x becomes 7x.
This gives us the simplified equation:
Now, the equation looks a bit cleaner and more manageable. But we're still not at our final answer. We need to isolate 'x' on one side of the equation. And that brings us to the crucial question: how do we get from here to Step 2? What mathematical operation allows us to move that constant term (-25) from the left side to the right side? The answer lies in understanding the properties of equality, which we'll explore in detail as we dissect Step 2.
The Heart of the Matter: Justifying Step 2
Now we arrive at the core of our exploration: understanding the justification for Step 2. We're moving from:
to
Notice what's changed: The '-25' has vanished from the left side, and a '+24' has appeared on the right side. How did that happen? This isn't just a random switcheroo; it's a carefully executed move based on a fundamental principle of algebra: the addition property of equality.
The addition property of equality states that you can add the same value to both sides of an equation without changing the equation's balance. Think of an equation like a perfectly balanced scale. If you add weight to one side, you must add the same weight to the other side to keep it balanced. Otherwise, the scale tips, and the equation becomes untrue.
In our case, we wanted to get rid of the '-25' on the left side. The opposite of subtracting 25 is adding 25. So, we added 25 to both sides of the equation. This is the key to understanding the justification. Let's see it in action:
On the left side, -25 and +25 cancel each other out, leaving us with just 7x. On the right side, -1 + 25 equals 24. And that's how we arrive at Step 2:
So, the answer to our initial question is clear: Step 2 is justified by the addition property of equality. This property is a cornerstone of algebraic manipulation, allowing us to isolate variables and solve for unknowns. Without it, we'd be stuck! It's crucial to internalize this concept, as it's used repeatedly in solving various types of equations. Remember, the addition property of equality is your friend in the world of algebra! It helps you maintain balance and move closer to the solution with every step.
Why Not Subtraction?
You might be wondering, "If there's an addition property of equality, is there a subtraction property too?" The answer is yes, and it's essentially the same concept in reverse. The subtraction property of equality states that you can subtract the same value from both sides of an equation without changing its balance. It's a powerful tool for isolating variables and simplifying equations, just like its addition counterpart.
So, why did we specifically choose addition to justify Step 2 in our problem? The answer lies in what we were trying to achieve: eliminating the '-25' term on the left side of the equation. To do this, we needed to perform the opposite operation. Since the term was being subtracted, we needed to add to cancel it out.
Think of it like this: If you have a debt of $25, you need to add $25 to your account to bring the balance back to zero. Similarly, in our equation, we needed to add 25 to both sides to eliminate the '-25' term.
While we could technically frame the move as subtracting a negative number (subtracting -25 is the same as adding 25), it's clearer and more direct to use the addition property of equality in this case. It aligns perfectly with the action we took: adding 25 to both sides. This clarity in understanding is crucial for preventing confusion and ensuring accuracy in your algebraic manipulations. So, while both addition and subtraction properties are valuable tools, we chose the one that best fit our specific goal in Step 2: eliminating a subtracted term by adding its positive counterpart.
Beyond Addition: Other Properties of Equality
The addition and subtraction properties of equality are just the tip of the iceberg! There are other essential properties that allow us to manipulate equations and solve for unknowns. Understanding these properties is crucial for becoming a confident and skilled problem-solver in algebra. Let's take a quick look at some other key players:
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The Multiplication Property of Equality: This property states that you can multiply both sides of an equation by the same non-zero value without changing the equation's balance. It's incredibly useful for getting rid of fractions or coefficients that are dividing the variable.
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The Division Property of Equality: Similar to multiplication, this property allows you to divide both sides of an equation by the same non-zero value without affecting the balance. It's often used to isolate a variable when it's being multiplied by a coefficient.
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The Distributive Property: While not strictly a "property of equality," the distributive property is a fundamental tool for simplifying equations. It allows you to multiply a term across a sum or difference within parentheses. For example, a(b + c) = ab + ac. This is key for expanding expressions and combining like terms.
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The Substitution Property: This property states that if two expressions are equal, you can substitute one for the other in any equation or expression. This is particularly useful when dealing with systems of equations or when you know the value of a specific variable.
These properties, along with the addition and subtraction properties, form the bedrock of algebraic manipulation. They provide the rules of the game, allowing us to transform equations while maintaining their truth. Mastering these properties will not only help you solve equations more efficiently but will also deepen your understanding of the underlying mathematical principles. So, embrace these tools, practice using them, and watch your algebraic skills soar! Remember, each property has its specific purpose, and knowing when and how to apply them is the key to success in algebra.
Putting It All Together: A Step-by-Step Recap
Okay, guys, let's consolidate what we've learned and walk through the entire process again, step by step. This will solidify your understanding and give you a clear picture of how the properties of equality work together to solve equations. We started with the equation:
Step 1: Combine Like Terms
The first thing we did was simplify the left side by combining the 'x' terms:
This gave us:
This step is based on the distributive property and the commutative property of addition, which allow us to rearrange and combine terms that are similar.
Step 2: Apply the Addition Property of Equality
This was the heart of our discussion! We wanted to get rid of the '-25' on the left side, so we added 25 to both sides of the equation:
This simplifies to:
This step is justified by the addition property of equality, which ensures that we maintain the balance of the equation.
Next Steps (Beyond Our Focus Question):
While our main goal was to understand Step 2, let's briefly outline the remaining steps to solve for 'x':
Step 3: Apply the Subtraction Property of Equality
Subtract 4x from both sides:
This gives us:
Step 4: Apply the Division Property of Equality
Divide both sides by 3:
This gives us the final answer:
So, there you have it! We've not only identified the justification for Step 2 but also seen how it fits into the larger process of solving an equation. Remember, each step is a deliberate move based on the properties of equality, and understanding these properties is key to mastering algebra.
Final Thoughts: Mastering the Fundamentals
Guys, understanding the justification for each step in solving an equation is way more important than just memorizing the steps themselves. It's like learning the rules of a game – you can't play well if you don't know why certain moves are allowed. In algebra, the properties of equality are our rules, and they ensure that our solutions are valid and accurate.
We've seen how the addition property of equality played a crucial role in Step 2, allowing us to isolate terms and simplify the equation. But we've also touched on other important properties like subtraction, multiplication, and division. These properties, along with the distributive property, are the fundamental tools in our algebraic toolkit.
By focusing on the why behind each step, you'll develop a deeper understanding of algebra and become a more confident problem-solver. You'll be able to tackle complex equations with ease, knowing that you're not just following a recipe, but applying sound mathematical principles. This understanding will also serve you well in more advanced math courses, where these concepts are built upon.
So, keep asking questions, keep exploring the reasons behind the rules, and keep practicing. The more you understand the fundamentals, the more powerful your mathematical skills will become. And remember, algebra isn't just about finding the right answer; it's about understanding the journey and the logic that gets you there! Go get 'em, mathletes!
