Finding The Range Of A Linear Equation

Hey guys! Today, we're diving into a cool math problem that's all about understanding ranges and how they relate to equations. Specifically, we're going to figure out the range of the relation 4x + y = 3, given the domain {-2, 1, 5}. It's like a treasure hunt, but instead of gold, we're finding numbers! This problem is a classic example of how to work with linear equations and understand their behavior. So, grab your thinking caps, and let's get started. We'll break down the question step by step, so even if you're new to this, you'll be acing it in no time. The key here is to understand the relationship between the domain (the input values) and the range (the output values). So, what does this actually mean? Let's clarify some core concepts before jumping into the solution. It means we want to find the possible y-values by substituting the given x-values into the equation. So, the question gives us the domain, the set of x-values we can use. Then, it provides the equation. Our job is to find the corresponding y-values for each x-value in the domain. These y-values will make up our range. Remember, the domain is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values) that result from plugging in those inputs into our function or equation. It's like a machine: you put something in (x-value), and you get something out (y-value). Pretty neat, right? The equation 4x + y = 3 is a linear equation. Linear equations are those that, when graphed, make a straight line. They have a constant rate of change, which means the y-value changes at a consistent rate for every change in the x-value. That makes these equations predictable and relatively easy to solve for specific values, given the domain, like in our problem. The approach involves substituting each x-value from the domain into the equation and then solving for y. This way, we’ll systematically find the corresponding y-value for each given x-value, which collectively form the range. This process ensures we cover all possible outputs for the given inputs and find the correct range. Now, let’s get into the step-by-step solution to unravel the answer. This is the fun part, so keep reading!

Step-by-Step Solution

Alright, let’s get down to business and figure this out. We have our domain: -2, 1, 5}. We also have our equation 4x + y = 3. Our goal is to find the range. To do this, we're going to substitute each value from the domain into the equation and solve for y. It’s like a mini-adventure where we get to explore how each input transforms into an output. Let's start with x = -2. Plugging this into our equation, we get 4(-2) + y = 3. Simplifying this, we get -8 + y = 3. Now, to isolate y, we add 8 to both sides, which gives us y = 11. So, when x = -2, y = 11. This means that the point (-2, 11) is a point on our line. Now let’s move on to the next value in our domain, which is x = 1. Substituting x = 1 into the equation, we get 4(1) + y = 3. This simplifies to 4 + y = 3. To solve for y, we subtract 4 from both sides. That gives us y = -1. So, when x = 1, y = -1. Awesome! This means that the point (1, -1) is also on our line. Only one more value in the domain to go! Last but not least, let’s plug in the final value from the domain, which is x = 5. Substituting this into the equation, we get 4(5) + y = 3. This simplifies to 20 + y = 3. To isolate y, we subtract 20 from both sides, which gives us y = -17. So, when x = 5, y = -17. Great job! This means the point (5, -17) is also on our line. We have found the corresponding y-values for each x-value in the domain. Therefore, the range is the set of all y-values we found. That is: {11, -1, -17. This is the set of output values that result from the given input values and the equation. That’s all there is to it! We have successfully determined the range for the given domain and equation. Let's compare the range we calculated to the options provided. It’s like we have our own treasure map, and now we know where the treasure (the correct answer) is buried! This step helps us confirm our findings and select the right option. This is a crucial step to ensure the accuracy of our solutions, which is a great exercise for your problem-solving skills.

Finding the Correct Answer

We did it, guys! We have all the pieces of the puzzle and are ready to unveil the answer. The question asked for the range, and we calculated it to be {11, -1, -17}. Now, we'll go back to the multiple-choice options and see which one matches our calculated range. Let’s review the answer choices:

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

By carefully comparing our range {11, -1, -17} to the options, it’s clear that option D is the correct one. It's an exact match! So, we know we did our calculations correctly. Always remember to check your work; it's a great habit. It helps catch any tiny mistakes. It's always a good idea to double-check your calculations, especially when dealing with math problems. This also boosts your confidence because you know you got the right answer. Yay, we found it! Choosing the correct answer isn't just about matching the numbers, it is about understanding the logic and the steps you took to get there. Understanding why an option is correct is just as important as knowing the right answer. Now we have finished solving the problem, and we know our range is {11, -1, -17}. Therefore, the correct answer is D. {11, -1, -17}. That’s all for today, everyone! We hope you enjoyed going through this problem. Keep practicing, and you will be math masters in no time! Remember, every problem is a chance to learn something new. Keep up the great work! Always remember to review your work and the concepts you learned. If you have any questions, feel free to ask. Cheers!