Solve Y - 2x = -3: Finding (x, Y) Values

Aug 1, 2025 by ADMIN 41 views

Decoding the Equation: Finding the Values that Make y - 2x = -3 True

Hey guys! Let's dive into this math problem together. We're on a mission to figure out which values of x and y make the equation y - 2x = -3 true. It's like a puzzle, and we're the detectives! We'll break it down step by step and explore different approaches to find the solutions. So, grab your thinking caps, and let's get started!

Understanding the Equation: A Linear Relationship

At its core, the equation y - 2x = -3 represents a linear relationship between x and y. This means that if we were to plot all the possible (x, y) pairs that satisfy this equation on a graph, we'd get a straight line. Pretty neat, huh? The equation is in a form that we can easily manipulate to understand this relationship better. To really get a feel for what's going on, let's rearrange the equation to solve for y. This will give us a clearer picture of how y changes as x changes. By isolating y, we can see y as a function of x, which is super helpful for finding solutions. So, let’s do some algebraic magic and get y all by itself on one side of the equation. Remember, our goal is to rewrite the equation in the slope-intercept form (y = mx + b), which is like the secret decoder ring for linear equations. Once we have it in this form, we can easily identify the slope and y-intercept, giving us a ton of insight into the line represented by the equation. Thinking about the equation graphically is a powerful way to understand the infinite solutions it holds. Each point on the line represents an (x, y) pair that makes the equation true. By understanding the slope and y-intercept, we can quickly visualize the line and identify potential solutions. This visual approach can be a great way to check our algebraic solutions and ensure they make sense in the context of the graph.

Solving for y: Unveiling the Slope-Intercept Form

To solve for y, we simply add 2x to both sides of the equation:

y - 2x + 2x = -3 + 2x

This simplifies to:

y = 2x - 3

Now, we have the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. In this case, the slope (m) is 2, and the y-intercept (b) is -3. This tells us that for every increase of 1 in x, y increases by 2, and the line crosses the y-axis at -3. Understanding the slope and y-intercept is key to finding solutions. The slope tells us the rate at which y changes with respect to x, and the y-intercept gives us a starting point on the y-axis. With this information, we can start plugging in values for x and see what y values satisfy the equation. This is a fun way to explore the relationship between x and y and discover the infinite solutions that exist. Remember, every point on the line y = 2x - 3 is a solution to the original equation. So, by understanding the slope and y-intercept, we've unlocked a powerful tool for finding these solutions. We can also use this form to easily generate a table of values, which can be helpful for visualizing the line and identifying specific solutions that fit our needs.

Finding Solutions: Plugging in Values for x

Now that we have the equation in the form y = 2x - 3, we can easily find solutions by choosing values for x and plugging them into the equation to find the corresponding y values. This is like having a recipe – we plug in the x ingredient, and the equation tells us the y result! Let's try a few examples:

If x = 0, then y = 2(0) - 3 = -3. So, (x, y) = (0, -3) is a solution.
If x = 1, then y = 2(1) - 3 = -1. So, (x, y) = (1, -1) is a solution.
If x = 2, then y = 2(2) - 3 = 1. So, (x, y) = (2, 1) is a solution.

We can continue this process indefinitely, plugging in any value for x and finding a corresponding y value. This demonstrates that there are infinitely many solutions to this equation! Each (x, y) pair we find represents a point on the line y = 2x - 3. This method of substituting values for x is a powerful way to generate solutions and get a sense of the relationship between x and y. We can also use this method to check if a given (x, y) pair is a solution. Simply plug the values into the equation and see if it holds true. If the equation balances, then the point is a solution; otherwise, it's not. This technique is essential for verifying our solutions and ensuring accuracy.

Evaluating the Options: Which Solutions Fit?

Now, let's take a look at the options provided and see which ones match the solutions we've found.

Option A: Some values of (x, y) are (0, -3), (1, -1), (2, 1), etc.
Option B: Some values of (x, y) are (0, 3), (1, 1), (2, -1), etc.
Option C: Some values of (x, y) are (0, -2), (-1, -1), (-2, 1), etc.

We already calculated that (0, -3), (1, -1), and (2, 1) are solutions. This perfectly matches Option A. Let's double-check the other options to be sure.

For Option B, if x = 0, then y = 2(0) - 3 = -3, not 3. So, Option B is incorrect.

For Option C, if x = 0, then y = 2(0) - 3 = -3, not -2. So, Option C is also incorrect.

Therefore, the correct answer is Option A. We've successfully identified the values of (x, y) that satisfy the equation! This process of evaluating the options by comparing them to our calculated solutions is a crucial step in problem-solving. It ensures that we're not just guessing but rather making an informed decision based on our understanding of the equation. By systematically checking each option, we can confidently arrive at the correct answer. This approach is particularly useful in multiple-choice questions, where the options provide a set of possible solutions to choose from.

Conclusion: Unlocking the Secrets of Linear Equations

So, there you have it, guys! We've successfully cracked the code and found the values of (x, y) that make the equation y - 2x = -3 true. We learned that this equation represents a linear relationship, and by rearranging it into slope-intercept form, we could easily find solutions by plugging in values for x. This whole process highlights the power of algebra in helping us understand and solve mathematical problems. By manipulating equations and applying our knowledge of linear relationships, we can unlock the secrets hidden within these mathematical expressions. Remember, math isn't just about numbers and formulas; it's about problem-solving and critical thinking. The skills we've used in this example, such as rearranging equations, substituting values, and evaluating options, are valuable tools that can be applied to a wide range of problems. So, keep practicing, keep exploring, and keep unlocking the amazing world of mathematics!

We also saw how important it is to check our work and make sure our answers make sense. By verifying the solutions in Option A and ruling out Options B and C, we gained confidence in our final answer. This attention to detail is crucial in mathematics and in life in general. Always double-check your work, and don't be afraid to ask for help or clarification if you're unsure about something. Learning is a collaborative process, and we all benefit from sharing our knowledge and supporting each other.

Remember, the world of linear equations is vast and fascinating. There's so much more to explore, from graphing lines to solving systems of equations. But the fundamental principles we've discussed here – understanding slope-intercept form, substituting values, and checking our work – will serve as a solid foundation for your mathematical journey. So, keep up the great work, and never stop learning! You've got this!