Direct Variation: Find Y When X = 7
Hey everyone! Today, we're diving into a classic math problem involving direct variation. This concept is super important in algebra and pops up in various real-world scenarios. So, let's break it down and make sure we understand it inside and out.
What is Direct Variation?
Before we jump into solving the problem, let's quickly recap what direct variation actually means. Two variables, say x and y, are said to vary directly if they are related by a constant multiple. In simpler terms, as x increases, y increases proportionally, and vice versa. This relationship can be expressed mathematically as:
where k is the constant of variation. This constant k is the key to understanding and solving direct variation problems. It tells us the exact relationship between x and y. To find k, we simply divide y by x for any corresponding pair of values.
Why is understanding direct variation important? Well, it's used to model all sorts of things! Think about the relationship between the number of hours you work and the amount you get paid, or the distance a car travels and the amount of fuel it consumes (assuming constant speed). Direct variation helps us make predictions and understand how quantities change together. It's a foundational concept, guys, so nailing it down is super beneficial.
Identifying Direct Variation in a Table
The problem presents us with a table of x and y values. To confirm that we're dealing with direct variation, we need to check if the ratio of y to x is consistent across all data points. If it is, we've found our constant of variation, k.
Looking at the provided table:
| x | 2 | 4 | 6 | 1 |
|---|---|---|---|---|
| y | 12 | 24 | 36 | 6 |
Let's calculate the ratio y/x for each pair:
- For x = 2, y = 12: 12 / 2 = 6
- For x = 4, y = 24: 24 / 4 = 6
- For x = 6, y = 36: 36 / 6 = 6
- For x = 1, y = 6: 6 / 1 = 6
See that? The ratio is 6 in every case! This confirms that y varies directly with x, and our constant of variation, k, is 6. We can now confidently say that the relationship between x and y is defined by the equation:
This equation is the magic formula that will help us solve for any value of y given a corresponding value of x.
Real-World Examples of Direct Variation
To really solidify our understanding, let's consider some real-world examples where direct variation comes into play:
- Earning Wages: The amount you earn is directly proportional to the number of hours you work, assuming you have a fixed hourly rate. If you earn $15 per hour, your earnings (y) can be represented as y = 15x, where x is the number of hours worked. This perfectly illustrates direct variation!
- Cooking Recipes: When you're scaling up or down a recipe, the quantities of ingredients vary directly with the number of servings you want to make. If a recipe for 4 servings calls for 1 cup of flour, then a recipe for 8 servings will need 2 cups of flour, and so on. It's a direct variation relationship!
- Distance and Time (at Constant Speed): If you're traveling at a constant speed, the distance you cover varies directly with the time you spend traveling. For instance, if you're driving at 60 miles per hour, the distance (y) you travel is y = 60x, where x is the time in hours. Direct variation in action!
These examples help us appreciate how direct variation isn't just a math concept; it's a fundamental relationship that governs many aspects of our lives. Recognizing these relationships makes problem-solving much more intuitive.
Solving for y When x is 7
Alright, now that we've established the equation $y = 6x$, we can finally answer the question: What is the value of y when x is 7? This is where the beauty of direct variation shines – it's a simple substitution!
We know that y = 6x, and we're given that x = 7. So, we just plug in 7 for x in our equation:
Now, it's just basic multiplication:
And that's it! When x is 7, y is 42. We've successfully used the direct variation relationship to find the value of y.
Why This Method Works
The reason this method works so seamlessly lies in the definition of direct variation. The constant of variation, k, ensures that the ratio between y and x remains the same, no matter what the value of x is. We found that k = 6, meaning that y is always 6 times x. Therefore, if we know x, we can always find y by multiplying it by 6. It's a direct and predictable relationship, guys, making it super handy for solving problems like this.
This method allows for quick and accurate calculations, especially in scenarios where you might not have a complete table of values. Understanding the underlying principle of direct variation empowers you to solve for any unknown variable efficiently.
Answering the Question
So, to answer the original question directly: Given that y varies directly with x in the table provided, the value of y when x is 7 is 42. We arrived at this answer by first identifying the constant of variation, k, from the table, then establishing the equation y = 6x, and finally substituting x = 7 into the equation.
The Importance of Showing Your Work
While arriving at the correct answer is crucial, showing your work is equally important, especially in mathematical problem-solving. Demonstrating the steps you took to reach the solution not only helps you double-check your work for errors but also provides a clear understanding of your thought process. In educational settings, showing your work often earns you partial credit even if the final answer is incorrect, as it showcases your grasp of the underlying concepts.
In this case, showing your work would involve:
- Calculating the constant of variation (k) from the table.
- Writing the equation for direct variation (y = kx).
- Substituting the given value of x (7) into the equation.
- Performing the calculation to find y.
By presenting your solution in a clear and organized manner, you not only validate your answer but also enhance your problem-solving skills and communication abilities.
Conclusion: Mastering Direct Variation
Guys, we've successfully tackled a direct variation problem! We started by understanding the definition of direct variation, identified it in a table of values, calculated the constant of variation, and finally, used the equation to find the value of y when x is 7. Remember, the key to these problems is identifying that constant of variation – it unlocks everything!
Direct variation is a fundamental concept that builds the groundwork for more advanced mathematical topics. By understanding it thoroughly, you'll be well-equipped to tackle a wide range of problems, not just in math class, but also in real-life situations. Keep practicing, and you'll become a direct variation pro in no time!
If you found this explanation helpful, keep exploring other math concepts and challenging yourself. Math is a journey, and each problem you solve strengthens your understanding and skills. Keep up the great work, guys!
