Solving Equations And Geometry Problems Step-by-Step

Hey guys! Today, we're diving into the exciting world of math, tackling both algebraic equations and geometric problems. We'll break down the steps, explain the concepts, and make sure you understand how to solve these types of questions. So, grab your pencils, and let's get started!

Solving the Equation: (25 - p) / 6 = 3

Solving equations is a fundamental skill in algebra, and it's essential for understanding more advanced mathematical concepts. In this section, we'll walk through the process of solving the equation (25 - p) / 6 = 3 step-by-step, ensuring you grasp the underlying principles. To begin solving the equation (25 - p) / 6 = 3, our main goal is to isolate the variable 'p' on one side of the equation. This means we need to undo any operations that are being performed on 'p'. The first operation we see is division by 6. To undo this, we need to multiply both sides of the equation by 6. This keeps the equation balanced, which is crucial in solving equations. When we multiply both sides by 6, we get: (25 - p) / 6 * 6 = 3 * 6. This simplifies to 25 - p = 18. Now, we have a simpler equation to work with. The next step is to isolate 'p' further. We have '25 - p', which means 'p' is being subtracted from 25. To isolate 'p', we can subtract 25 from both sides of the equation. This gives us: 25 - p - 25 = 18 - 25. Simplifying this, we get -p = -7. Notice that 'p' is still not completely isolated; it has a negative sign in front of it. To get 'p' by itself, we need to multiply both sides of the equation by -1. This will change the sign of both sides. So, we have -p * -1 = -7 * -1. This simplifies to p = 7. Therefore, the solution to the equation (25 - p) / 6 = 3 is p = 7. To double-check our answer, we can substitute p = 7 back into the original equation and see if it holds true. Substituting p = 7 into (25 - p) / 6 = 3, we get (25 - 7) / 6 = 3. Simplifying the numerator, we have 18 / 6 = 3. And indeed, 18 / 6 equals 3, so our solution is correct. In summary, we solved the equation (25 - p) / 6 = 3 by first multiplying both sides by 6 to eliminate the division. Then, we subtracted 25 from both sides to further isolate 'p'. Finally, we multiplied both sides by -1 to solve for 'p'. By following these steps, we found that p = 7 is the solution to the equation. Remember, the key to solving equations is to perform the same operations on both sides to maintain balance and isolate the variable. This method can be applied to a wide range of algebraic equations, making it a fundamental skill in mathematics. Great job, guys! You've successfully solved your first equation of the day. Let's move on to the geometry problem now!

Finding the Area of a Rectangle

Now, let's switch gears and tackle a geometry problem. We're given the width of a rectangle and need to find its area. But wait, there seems to be some missing information! Let's break down the problem and see what we can figure out.

The question states: “Ширина прямоугольника 4 см. Чему равна площадь прямоугольника, если его…”. It seems like the sentence is incomplete, which makes it impossible to find a definitive answer for the area. To calculate the area of a rectangle, we need both its width and its length. The formula for the area of a rectangle is: Area = Length × Width. We are given the width, which is 4 cm, but the length is missing from the problem statement. Without the length, we cannot calculate the area of the rectangle. The area could be any value depending on the length. For example, if the length was 5 cm, the area would be 4 cm × 5 cm = 20 square centimeters. If the length was 10 cm, the area would be 4 cm × 10 cm = 40 square centimeters, and so on. Because the length is not provided, we can only express the area in terms of the length. If we denote the length as 'l' centimeters, then the area 'A' can be expressed as: A = 4 × l square centimeters. This means the area of the rectangle is 4 times its length. To find a specific numerical value for the area, we would need the length. So, without additional information, we can't give a single numerical answer for the area. Instead, we acknowledge that the area depends directly on the length and can be calculated once the length is known. In real-world scenarios, always ensure you have all the necessary information before attempting to solve a problem. Missing data can lead to incomplete or incorrect solutions. In this case, the problem highlights the importance of having both the length and width to accurately calculate the area of a rectangle. When faced with such situations, it’s crucial to identify what information is missing and determine if it can be obtained or estimated. Sometimes, you might need to make assumptions or use additional information from the context to proceed with the problem. However, it's important to state any assumptions you make clearly. In conclusion, while we cannot provide a specific numerical answer for the area of the rectangle with just the width given, we understand the relationship between the length and the area. The area can be calculated using the formula Area = 4 × l, where 'l' is the length of the rectangle in centimeters. We need the length to find the exact area. Remember guys, always double-check your problem statements to make sure you have all the necessary information. It's a crucial step in problem-solving! Now that we've analyzed this geometry problem, let's recap what we've learned today.

Key Takeaways

In today's session, we've covered two important mathematical concepts: solving algebraic equations and finding the area of a rectangle. For solving the equation (25 - p) / 6 = 3, we learned the importance of isolating the variable by performing the same operations on both sides of the equation. We systematically worked through the steps: multiplying by 6, subtracting 25, and multiplying by -1 to find that p = 7. This approach is fundamental in algebra and can be applied to a variety of equations. Remember, the key is to maintain balance by performing the same operation on both sides, ensuring the equation remains equivalent throughout the solving process. Checking your solution by substituting it back into the original equation is also a crucial step to verify its correctness. Moving on to the geometry problem, we encountered an incomplete statement regarding the area of a rectangle. We were given the width but lacked the length, which is necessary to calculate the area. This situation highlighted the significance of having all the required information before attempting to solve a problem. We reiterated the formula for the area of a rectangle (Area = Length × Width) and explained that without the length, we could only express the area in terms of the unknown length 'l', as A = 4 × l. This exercise underscored the importance of carefully analyzing problem statements and identifying any missing information. It also demonstrated how mathematical expressions can represent relationships between variables even when specific values are not available. In summary, today's session emphasized the methodical approach to solving algebraic equations and the critical importance of complete information in geometry problems. These are foundational concepts in mathematics, and understanding them is crucial for further learning. By practicing these skills, you'll become more confident in your ability to tackle mathematical challenges. Keep practicing, and you'll master these concepts in no time. You've done a great job today, guys! Remember, math is like building blocks – each concept builds upon the previous one. So keep learning and keep growing!

This was a fantastic session, and I'm proud of how you tackled these problems. Keep up the great work, and I'll see you in the next one for more mathematical adventures!