Finding The Range Of A Relation 4x+y=3 With Domain {-2, 1, 5}

Hey everyone! Today, we're diving into a fascinating math problem that involves finding the range of a relation. It's a concept that might sound a bit intimidating at first, but trust me, it's super manageable once you break it down. We'll walk through the solution step-by-step, making sure you understand the 'why' behind each move. So, grab your thinking caps, and let's get started!

The Problem: Unveiling the Range

Let's kick things off by stating the problem clearly. We're given a relation defined by the equation 4x + y = 3. The domain, which is the set of possible x-values, is {-2, 1, 5}. Our mission, should we choose to accept it, is to determine the range – the set of all possible y-values that result from plugging in our domain values into the equation. To put it simply, we have a set of 'x' values, a mathematical rule (our equation), and we want to find out what 'y' values we get out of it. So, how do we tackle this? The core idea here is substitution. We'll take each x-value from our domain and substitute it into the equation. This will give us a corresponding y-value. Once we've done this for all x-values in the domain, we'll have our range. This is a fundamental concept in algebra and is used extensively in various mathematical fields. Understanding how to find the range of a relation is crucial for grasping functions, graphs, and various other mathematical concepts. It's like learning the alphabet before you can read; this skill is a building block for more advanced topics. Before we jump into the calculations, let's take a moment to understand why this process works. The equation 4x + y = 3 defines a relationship between x and y. For every value of x, there's a corresponding value of y that makes the equation true. The domain limits the possible x-values we can consider. So, by substituting each x-value, we're essentially finding the y-values that 'fit' the equation within the given domain. This process is not just about plugging in numbers; it's about understanding the connection between variables and how equations define their relationship. It's like understanding how ingredients interact in a recipe; each ingredient (x-value) contributes to the final dish (y-value). Now, with this understanding in place, we're ready to roll up our sleeves and get calculating! We'll start with the first x-value in our domain and work our way through, carefully tracking the y-values we find. Remember, accuracy is key in math, so we'll double-check our work as we go. Let's dive in!

Step-by-Step Solution: Cracking the Code

Alright, let's get down to business and solve this step-by-step. Remember, our goal is to find the range by substituting each x-value from the domain into the equation 4x + y = 3. So, let's break it down:

1. Substituting x = -2

Our first x-value from the domain is -2. We're going to plug this into our equation and solve for y. This means replacing 'x' with '-2' in the equation 4x + y = 3. So, here we go: 4 * (-2) + y = 3. Now, let's simplify this. 4 multiplied by -2 gives us -8. So, our equation now looks like this: -8 + y = 3. We're trying to isolate 'y' to find its value. To do that, we need to get rid of the '-8' on the left side of the equation. The opposite of subtracting 8 is adding 8, so we'll add 8 to both sides of the equation. This keeps the equation balanced, which is a golden rule in algebra. So, adding 8 to both sides gives us: -8 + y + 8 = 3 + 8. The -8 and +8 on the left side cancel each other out, leaving us with just 'y'. On the right side, 3 + 8 equals 11. So, we have our first y-value: y = 11. This means that when x is -2, y is 11. This is one piece of the puzzle in figuring out the range. We've successfully found the y-value corresponding to the first x-value in our domain. But we're not done yet! We still have two more x-values to substitute and solve for. Each x-value will give us a different y-value, which will contribute to our final range. Remember, the range is the set of all possible y-values, so we need to find them all. Let's keep this momentum going and move on to the next x-value. The process will be similar – substitute, simplify, and solve for y. With each step, we're getting closer to unveiling the complete range of the relation. It's like we're detectives, and each calculation is a clue that helps us crack the case. So, let's sharpen our pencils, focus our minds, and move on to the next x-value!

2. Substituting x = 1

Now, let's tackle the second x-value in our domain, which is 1. Just like before, we're going to substitute this value into our equation 4x + y = 3 and solve for y. So, we replace 'x' with '1': 4 * (1) + y = 3. This simplifies to 4 + y = 3. Remember, our goal is to isolate 'y' to find its value. Currently, we have a '4' added to 'y' on the left side of the equation. To get rid of this '4', we need to do the opposite operation, which is subtraction. We'll subtract 4 from both sides of the equation to keep it balanced. This is a crucial step in solving equations – whatever we do to one side, we must do to the other. So, subtracting 4 from both sides gives us: 4 + y - 4 = 3 - 4. The '4' and '-4' on the left side cancel each other out, leaving us with just 'y'. On the right side, 3 - 4 equals -1. Therefore, we have our second y-value: y = -1. This means that when x is 1, y is -1. We're making progress! We've now found two y-values that belong to the range of our relation. Each y-value corresponds to a specific x-value in our domain. It's like we're building a bridge, and each calculation is a pillar that supports the bridge. We're one step closer to completing the bridge and understanding the full relationship between x and y. But we're not there yet. We still have one more x-value to substitute and solve for. This final calculation will give us the last piece of the puzzle and allow us to define the complete range. Let's keep our focus sharp and move on to the final substitution. We're in the home stretch now, so let's make sure we finish strong and accurately!

3. Substituting x = 5

Alright, it's time for the final substitution! Our last x-value from the domain is 5. We're going to follow the same procedure as before: plug this value into the equation 4x + y = 3 and solve for y. So, let's replace 'x' with '5': 4 * (5) + y = 3. This simplifies to 20 + y = 3. Now, we need to isolate 'y' to find its value. We have '20' added to 'y' on the left side of the equation. To get rid of this '20', we'll subtract 20 from both sides. Remember, maintaining balance is key! So, subtracting 20 from both sides gives us: 20 + y - 20 = 3 - 20. The '20' and '-20' on the left side cancel each other out, leaving us with just 'y'. On the right side, 3 - 20 equals -17. Therefore, we have our third and final y-value: y = -17. This means that when x is 5, y is -17. We've done it! We've successfully found the y-values corresponding to all the x-values in our domain. This was the last piece of the puzzle, and now we can finally assemble the complete picture of the range. We've gone through the process of substitution, simplification, and solving for y three times, and each time we've obtained a different y-value. These y-values together form the range of our relation. It's like we've collected all the ingredients for a recipe, and now we're ready to put them together to create the final dish. But before we celebrate, let's take a moment to organize our findings and present the range in a clear and concise way. This will ensure that our answer is not only correct but also easy to understand. So, let's gather our y-values and prepare to state the range!

The Range: Unveiled!

Okay, guys, after all that number crunching, we've finally arrived at the range! We found the following y-values:

• When x = -2, y = 11
• When x = 1, y = -1
• When x = 5, y = -17

So, the range is the set of these y-values: {11, -1, -17}. Now, let's take a look at the answer choices provided in the original problem.

• A. {-11, 1, 17}
• B. {11, -1, -17}
• C. {-1, 4, 8}
• D. {-5, 7, 23}

Comparing our calculated range {11, -1, -17} with the answer choices, we can clearly see that option B matches perfectly. Boom! We've nailed it! We've successfully determined the range of the relation for the given domain. It's like we've solved a puzzle, and the pieces have all fallen into place. We started with the equation and the domain, and through careful substitution and calculation, we've unveiled the range. This process not only gives us the answer but also strengthens our understanding of mathematical relationships and problem-solving techniques. It's a fantastic feeling to conquer a challenging problem, isn't it? We've demonstrated our ability to apply algebraic principles to find the range of a relation, and this skill will serve us well in future mathematical endeavors. But let's not stop here! Understanding the solution is just the first step. To truly master this concept, it's beneficial to reflect on the process and the underlying principles. This will solidify our knowledge and make us even more confident in tackling similar problems. So, let's take a moment to recap the key steps we took and the concepts we utilized.

Key Takeaways: Mastering the Concept

So, what did we learn today? Let's recap the key steps and concepts involved in finding the range of a relation:

1. Understanding the Domain and Range: The domain is the set of input values (x-values), and the range is the set of output values (y-values) that result from applying the relation's equation.
2. Substitution is Key: The core technique is to substitute each x-value from the domain into the equation.
3. Solving for y: After substituting, we solve the equation for y to find the corresponding y-value.
4. Collecting the y-values: The range is the set of all the y-values we obtain after substituting all x-values from the domain.
5. Double-Check Your Work: Accuracy is crucial in math, so always double-check your calculations to avoid errors.

This process might seem straightforward, but it's a fundamental concept that underpins many areas of mathematics. Think of it like learning the basic chords on a guitar; once you've mastered them, you can play a whole range of songs. Similarly, understanding how to find the range of a relation opens the door to understanding more complex functions and graphs. But it's not just about following the steps; it's about understanding why they work. When we substitute an x-value into the equation, we're essentially finding the y-value that 'fits' the equation. The equation defines a relationship between x and y, and the range is the set of all y-values that satisfy this relationship within the given domain. This understanding is crucial for applying this concept in different contexts. For example, in real-world scenarios, we might use this concept to model the relationship between different variables, such as the amount of time spent studying and the grade received on an exam. The domain would represent the possible hours spent studying, and the range would represent the possible grades. Understanding the range allows us to make predictions and draw conclusions about the relationship between these variables. So, keep practicing these steps, and don't be afraid to tackle different types of problems. The more you practice, the more comfortable and confident you'll become. And remember, math is not just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively. So, keep exploring, keep questioning, and keep learning!

Practice Makes Perfect: Test Your Skills!

Alright, folks, now that we've walked through the solution and recapped the key takeaways, it's time to put your newfound knowledge to the test! The best way to truly master a concept is through practice. So, let's try a similar problem to solidify your understanding. Imagine we have a different relation defined by the equation 2x - y = 5, and the domain is {0, 2, 4}. Can you find the range? I encourage you to grab a pen and paper and work through the problem yourself. Follow the same steps we used in the previous example: substitute each x-value from the domain into the equation, solve for y, and then collect all the y-values to form the range. Don't be afraid to make mistakes – that's how we learn! The important thing is to try and understand the process. And if you get stuck, don't worry! You can always refer back to the steps we outlined earlier. Once you've found the range, compare your answer with others or check online resources to see if you're on the right track. You can even create your own problems with different equations and domains to further challenge yourself. The more you practice, the more confident you'll become in your ability to find the range of a relation. And remember, math is like a muscle; the more you exercise it, the stronger it gets. So, keep practicing, keep challenging yourself, and keep growing your mathematical skills. And who knows, maybe one day you'll be the one explaining these concepts to others! Now, go forth and conquer that practice problem! I have faith in you. And remember, math is not just about getting the right answer; it's about the journey of learning and discovery. So, enjoy the process, and happy problem-solving!