Solving Famema SP Question 49 Intercepts And Distance Calculation

Hey guys! Today, we're diving deep into a fascinating math problem from Famema SP, specifically Question 49. This question beautifully combines concepts of intercepts, distance, and analytical geometry. So, buckle up and let's embark on this mathematical journey together! We'll break down the problem step-by-step, ensuring you not only understand the solution but also the underlying principles.

Understanding the Problem

Before we jump into the solution, let's make sure we fully grasp what the question is asking. The essence of the problem revolves around finding the intercepts of a line and then calculating the distance between those intercepts. Often, these types of problems will provide the equation of the line in some form. From the equation, we need to extract the information necessary to identify where the line crosses the x-axis (x-intercept) and the y-axis (y-intercept). Once we have these points, calculating the distance is a breeze using the distance formula. This question is a fantastic way to test your understanding of coordinate geometry and your ability to apply fundamental concepts in a practical scenario. It’s not just about plugging numbers into a formula; it’s about visualizing the geometry and connecting it with the algebraic representation. This is where the real magic of math happens, and it’s what makes problems like this so rewarding to solve.

Finding the Intercepts

Okay, let's get down to business and talk about finding those intercepts. The intercepts, as we mentioned, are the points where the line crosses the coordinate axes. The x-intercept is where the line intersects the x-axis, and at this point, the y-coordinate is always zero. Conversely, the y-intercept is where the line crosses the y-axis, and here, the x-coordinate is zero. This simple yet powerful concept is the key to unlocking this part of the problem.

To find the x-intercept, you substitute y = 0 into the equation of the line and solve for x. Think of it like this: you're forcing the equation to tell you what x value corresponds to the line being at the same vertical level as the x-axis. It’s a direct algebraic way of pinpointing that crossing point. Similarly, to find the y-intercept, you substitute x = 0 into the equation and solve for y. This time, you're asking the equation what y value matches the line's position when it's directly on the y-axis. These substitutions simplify the equation, often making it a straightforward process to isolate the remaining variable and find your intercept value. Remember, intercepts are points, so the final answer should be expressed as coordinate pairs, like (x, 0) for the x-intercept and (0, y) for the y-intercept. Mastering this technique is crucial not only for this problem but for a wide range of analytical geometry problems. It's a foundational skill that will serve you well in your mathematical journey.

Calculating the Distance

Now that we've successfully located our intercepts, the next step is to calculate the distance between them. This is where the distance formula comes into play, and it’s a fantastic tool to have in your math arsenal. The distance formula is derived from the Pythagorean theorem, which you might remember from geometry class. It provides a direct way to calculate the straight-line distance between any two points in a coordinate plane. The formula itself looks a bit intimidating at first, but it's quite manageable once you break it down.

If we have two points, let's call them (x1, y1) and (x2, y2), the distance d between them is given by: d = √((x2 - x1)² + (y2 - y1)²). Don't let the square root and the squares scare you! It's just a matter of plugging in the coordinates and following the order of operations. The essence of the formula is calculating the difference in the x-coordinates and the difference in the y-coordinates, squaring those differences, adding them together, and then taking the square root of the sum. This process effectively finds the length of the hypotenuse of a right triangle where the legs are the differences in the x and y coordinates. In our case, the points will be the x-intercept and the y-intercept that we calculated earlier. So, you'll simply substitute the coordinates of those intercepts into the formula, do the arithmetic, and voilà, you'll have the distance between them. This distance represents the length of the line segment connecting the two intercepts, and it’s the final piece of the puzzle for solving the problem.

Solving the Specific Problem (Example)

Alright, let's solidify our understanding by tackling a specific example that mirrors the kind of question you might encounter in Famema SP Question 49. Let's say we're given the equation of a line: 2x + 3y = 6. Our mission is to find the intercepts and then determine the distance between them. Remember, the process involves a systematic application of the concepts we've discussed, so let’s dive in.

First up, let's find the x-intercept. To do this, we substitute y = 0 into the equation: 2x + 3(0) = 6. This simplifies to 2x = 6. Dividing both sides by 2, we get x = 3. So, the x-intercept is the point (3, 0). We've successfully pinpointed where the line crosses the x-axis. Now, let’s move on to the y-intercept. We substitute x = 0 into the original equation: 2(0) + 3y = 6. This simplifies to 3y = 6. Dividing both sides by 3, we get y = 2. Therefore, the y-intercept is the point (0, 2). We now know where the line crosses the y-axis. With both intercepts in hand, we're ready for the final step: calculating the distance between them. We'll use the distance formula: d = √((x2 - x1)² + (y2 - y1)²). Let's plug in our points, (3, 0) and (0, 2). It doesn’t matter which point we call (x1, y1) and which we call (x2, y2), as long as we're consistent. So, let’s say (x1, y1) = (3, 0) and (x2, y2) = (0, 2). Substituting these values into the formula, we get: d = √((0 - 3)² + (2 - 0)²) = √((-3)² + (2)²) = √(9 + 4) = √13. Thus, the distance between the intercepts is √13 units. We've successfully navigated the problem from start to finish, finding the intercepts and calculating the distance between them. This example perfectly illustrates the steps involved in solving problems like Famema SP Question 49.

Common Mistakes and How to Avoid Them

It's super important to be aware of common mistakes that students often make when tackling problems like this. Knowing these pitfalls can help you steer clear of them and boost your chances of acing the question. One frequent error is mixing up the process for finding x and y-intercepts. Remember, to find the x-intercept, you set y = 0, and to find the y-intercept, you set x = 0. Getting these reversed is a classic blunder that can throw off your entire solution.

Another common mistake crops up when applying the distance formula. People sometimes forget to square the differences in the coordinates or might mess up the order of subtraction. It's crucial to follow the formula meticulously and double-check your calculations to avoid these errors. Sign errors are also a frequent culprit, especially when dealing with negative coordinates. Make sure you're paying close attention to the signs when substituting values into the formula and when simplifying the expressions. Furthermore, remember that the distance is always a non-negative value. If you end up with a negative distance, you know there's an error somewhere in your calculations. Finally, it’s easy to make arithmetic mistakes when simplifying the expressions under the square root. Take your time, write out each step clearly, and double-check your work to minimize the chances of these slip-ups. By being mindful of these common pitfalls and practicing careful calculation habits, you'll be well-equipped to tackle these types of problems with confidence and accuracy.

Practice Problems and Resources

To truly master this topic, practice is absolutely key. The more you work through problems, the more comfortable you'll become with the concepts and the techniques involved. Fortunately, there are tons of resources available to help you hone your skills. Start by revisiting similar problems from your textbook or class notes. Working through these examples again can reinforce your understanding of the process and help you identify any areas where you might need further clarification.

Online resources are also a goldmine for practice problems. Websites like Khan Academy, for instance, offer a wealth of exercises and video tutorials on coordinate geometry and the distance formula. You can also find practice questions on educational websites specifically designed for test preparation, such as those for the SAT or ACT, as these exams often include similar types of problems. Additionally, consider seeking out past exam papers or practice tests for Famema SP. These resources will give you a realistic sense of the types of questions you can expect and help you prepare effectively. Don't just focus on getting the right answer; take the time to understand the underlying concepts and the reasoning behind each step. Try different approaches to solving the same problem to develop a deeper understanding and problem-solving flexibility. By dedicating time to consistent practice and utilizing the available resources, you'll build a strong foundation in this area of mathematics and be well-prepared for any challenges that come your way.

Conclusion

So, there you have it, guys! We've thoroughly dissected Famema SP Question 49, focusing on intercepts and distance calculations. Remember, the key is to understand the underlying principles, practice diligently, and be mindful of common mistakes. With a solid grasp of these concepts and a bit of practice, you'll be well-equipped to tackle similar problems with confidence. Keep up the great work, and happy solving!