Hey guys! Let's dive into this interesting math problem together. We've got an equation that looks a bit complex at first glance, but don't worry, we'll break it down step by step. The heart of our task is to isolate y and express it in terms of x. This is a common type of problem in algebra, and mastering it can really boost your math skills. We aim to make this process as clear and easy as possible, so you can tackle similar problems with confidence. Think of it as a puzzle where x and y are pieces, and we need to rearrange the equation to see how they fit together. The equation we're dealing with is x = y + |-4| - |-5| + 6. It involves absolute values, addition, and subtraction, so we'll need to handle each part carefully. Before we jump into the solution, let’s quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of -4, written as |-4|, is 4, and the absolute value of -5, written as |-5|, is 5. Understanding this concept is crucial for solving the equation correctly. Now, let’s get started and find out what y equals in terms of x! We'll go through each step meticulously, ensuring you understand not just the 'how' but also the 'why' behind every operation. So, grab your calculators (though you might not need them for this one!), and let’s get solving!
Understanding the Equation x=y+|-4|-|-5|+6
The given equation, x = y + |-4| - |-5| + 6, might seem daunting at first, but it's actually quite manageable once we break it down. The key here is to understand each component and how they interact. Our ultimate goal is to isolate y, which means getting y by itself on one side of the equation. To do this, we need to simplify the equation and then perform some algebraic manipulations. Let's start by addressing the absolute values. As we discussed earlier, the absolute value of a number is its distance from zero. So, |-4| is 4, and |-5| is 5. Replacing these in our equation, we get x = y + 4 - 5 + 6. Now, the equation looks much simpler! We've eliminated the absolute value symbols and are left with basic arithmetic operations. The next step is to combine the constant terms on the right side of the equation. We have 4, -5, and 6. Adding these together, 4 minus 5 is -1, and -1 plus 6 is 5. So, our equation now looks like this: x = y + 5. We're getting closer to isolating y! This simplified form makes it much easier to see what we need to do next. We need to get rid of that +5 on the right side. Remember, whatever we do to one side of the equation, we must do to the other to keep things balanced. So, we'll subtract 5 from both sides. This step is crucial because it maintains the equality and moves us closer to our goal of having y alone on one side. By understanding these fundamental principles of algebra, we can confidently solve for y. Let’s move on to the next step and see how we finally isolate y and express it in terms of x.
Step-by-Step Solution to Find y
Okay, let's roll up our sleeves and get to the nitty-gritty of solving for y. We've already simplified our equation to x = y + 5. Now, it's just one step away from finding y! The key here is to isolate y on one side of the equation. To do this, we need to get rid of the +5 that's on the same side as y. The golden rule in algebra is that whatever you do to one side of the equation, you must do to the other. This ensures that the equation remains balanced. So, to eliminate the +5, we'll subtract 5 from both sides of the equation. This gives us: x - 5 = y + 5 - 5. On the right side, the +5 and -5 cancel each other out, leaving us with just y. On the left side, we have x - 5. So, our equation now looks like this: x - 5 = y. And there you have it! We've successfully isolated y. This equation tells us exactly what y is in terms of x. To make it crystal clear, we can simply rewrite the equation with y on the left side: y = x - 5. This is our final answer. We've expressed y as a function of x. This means that if you know the value of x, you can easily find the value of y by subtracting 5 from x. It’s like having a secret formula to unlock the value of y! This step-by-step process highlights the importance of algebraic manipulation and the balance of equations. By subtracting 5 from both sides, we maintained the equality and successfully isolated y. Now, let’s summarize our solution and make sure we've got a solid grasp on what we've achieved.
Expressing y in Terms of x The Final Answer
Alright, guys, let’s nail down our final answer and make sure we've got this concept down pat. We started with the equation x = y + |-4| - |-5| + 6 and went through a series of steps to isolate y. We simplified the absolute values, combined the constant terms, and then used algebraic manipulation to get y by itself. Remember, our goal was to express y in terms of x, meaning we wanted an equation that shows y equal to something that involves x. After all the simplifying and rearranging, we arrived at the equation y = x - 5. This is our final answer. It tells us that y is equal to x minus 5. In other words, to find the value of y, you simply take the value of x and subtract 5 from it. It’s that simple! This equation is a powerful tool because it allows us to easily find y for any given value of x. For example, if x is 10, then y would be 10 - 5, which is 5. If x is 0, then y would be 0 - 5, which is -5. You get the idea! Expressing one variable in terms of another is a fundamental skill in algebra and is used in many different contexts. It’s like having a recipe where you can adjust one ingredient (x) and see how it affects the final dish (y). So, to recap, the value of y in terms of x for the equation x = y + |-4| - |-5| + 6 is y = x - 5. We’ve successfully solved the problem and expressed y in terms of x. Now, let's wrap things up with a summary of what we've learned and how you can apply these skills to other problems.
Conclusion Mastering Algebraic Manipulations
So, guys, we've reached the end of our journey through this algebraic puzzle! We started with the equation x = y + |-4| - |-5| + 6 and, through careful simplification and algebraic manipulation, we successfully expressed y in terms of x. Our final answer is y = x - 5. This means that y is equal to x minus 5. This exercise wasn't just about finding the answer; it was about understanding the process of solving algebraic equations. We learned how to simplify expressions by dealing with absolute values and combining like terms. We also practiced the crucial skill of isolating a variable by performing the same operations on both sides of the equation. This is a fundamental concept in algebra and is essential for solving more complex problems. The ability to express one variable in terms of another is a powerful tool that you'll use in many areas of mathematics and beyond. It allows you to see the relationship between variables and how they affect each other. Think of it like understanding the gears in a machine – you can see how one gear turning affects the others. Now that you've mastered this problem, you're well-equipped to tackle similar challenges. Remember to always break down complex equations into smaller, manageable steps. Simplify where you can, and don't forget the golden rule of algebra: whatever you do to one side of the equation, you must do to the other. Keep practicing, and you'll become a pro at algebraic manipulations in no time! And that's a wrap! We hope you found this explanation helpful and that you feel more confident in your algebra skills. Keep up the great work, and we'll see you next time with another math challenge!