Direct Variation: Solve For Y

When we talk about direct variation in mathematics, we're describing a relationship between two variables where one variable is a constant multiple of the other. Essentially, as one variable increases, the other variable increases proportionally, and as one decreases, the other decreases proportionally. The core idea is that their ratio remains constant. You can express this relationship with a simple equation: y = kx, where 'y' is one variable, 'x' is the other variable, and 'k' is the constant of variation. This constant 'k' is super important because it tells us the specific rate at which 'y' changes with respect to 'x'. Finding this constant is usually the first step in solving any direct variation problem. Once you know 'k', you can easily find the value of one variable if you know the other, or vice versa. It’s like having a secret code that unlocks the relationship between them! Understanding direct variation is fundamental in many areas of math and science, from physics where you might see force varying directly with acceleration (F=ma, where 'm' is the constant mass), to economics where supply might vary directly with price.

Let's dive into a specific scenario to really nail this concept down. We're given that 'y' varies directly as 'x'. This means our trusty equation y = kx is our starting point. We're also provided with a specific pair of values: y is 6 when x is 72. This is our key to unlocking the constant of variation, 'k'. We can plug these numbers directly into our equation: 6 = k * 72. To find 'k', we just need to isolate it. We can do this by dividing both sides of the equation by 72. So, k = 6 / 72. Now, we can simplify this fraction. Both 6 and 72 are divisible by 6. 6 divided by 6 is 1, and 72 divided by 6 is 12. Therefore, our constant of variation, k, is 1/12. This means that 'y' is always 1/12th of 'x' in this particular relationship. This fraction might seem small, but it precisely defines how 'y' and 'x' are connected. Having found 'k', we've essentially determined the rule for this specific direct variation. This constant 'k' is the engine that drives the relationship between our variables. It’s the unchanging factor that holds the proportion true, no matter what values 'x' and 'y' take on, as long as they adhere to this direct variation rule. It's a critical piece of the puzzle, and once it's in place, the rest of the problem becomes much more straightforward.

Now that we've successfully determined the constant of variation, k = 1/12, we can tackle the second part of our problem. The question asks: what is the value of y when x is 8? We already have our direct variation equation, y = kx, and we know the value of 'k'. So, we simply substitute our known values into the equation: y = (1/12) * 8. To solve for 'y', we multiply 1/12 by 8. Multiplying a fraction by a whole number is straightforward: you multiply the numerator of the fraction by the whole number and keep the denominator the same. So, y = 8/12. Just like before, we should simplify this fraction to its lowest terms. Both 8 and 12 are divisible by 4. 8 divided by 4 is 2, and 12 divided by 4 is 3. Therefore, the value of y is 2/3 when x is 8. This result means that for this specific direct variation, when the 'x' value is 8, the corresponding 'y' value will be 2/3. This demonstrates the predictive power of understanding direct variation. By establishing the relationship (y=kx) and finding the constant (k), we can accurately determine unknown values within that relationship. It’s a powerful tool for understanding how quantities change together in a proportional manner. The simplicity of the formula belies its utility in modeling real-world phenomena where proportionality plays a significant role, making it a cornerstone concept in mathematical problem-solving.

Understanding Direct Variation in Broader Contexts

Direct variation isn't just an abstract mathematical concept; it's a principle that appears frequently in the real world, helping us understand and predict how different quantities relate to each other. Think about a recipe: if you want to double the batch, you double all the ingredients. The amount of flour needed varies directly with the number of servings you want to make. If one serving needs 1 cup of flour, two servings need 2 cups, and so on. The constant of variation here is the amount of flour per serving. In physics, Ohm's Law is a classic example of direct variation. It states that the voltage (V) across a resistor is directly proportional to the current (I) flowing through it, with the resistance (R) being the constant of proportionality (V = IR). If you increase the current, the voltage increases proportionally, assuming the resistance stays the same. Similarly, in finance, the simple interest earned on an investment can vary directly with the principal amount invested, assuming the interest rate and time period are constant. The more money you invest, the more interest you earn, and the relationship is linear. Even something as simple as buying apples at a market demonstrates direct variation. If apples cost $0.50 each, the total cost of the apples you buy varies directly with the number of apples you purchase. The price per apple ($0.50) is your constant of variation. The formula y = kx is a universal template for these proportional relationships. Recognizing these patterns allows us to make estimations, plan accordingly, and understand the underlying mechanisms of many everyday phenomena. It's the mathematical language that describes how things scale together consistently. This understanding empowers us to solve problems that involve proportional reasoning, whether in a classroom setting or in practical applications, making mathematics a valuable tool for navigating the world around us.

Key Takeaways for Direct Variation Problems

To successfully navigate problems involving direct variation, it's essential to remember a few core principles. First and foremost, always start by recognizing the keyword phrases that indicate direct variation, such as "varies directly as," "is proportional to," or "is a function of." These phrases signal that the relationship between the variables can be expressed as y = kx. The next crucial step is to identify the given pair of values for 'x' and 'y'. These values are your tools for calculating the constant of variation, k. By substituting these known values into the equation y = kx, you can then solve for 'k' by isolating it, typically through division. Once you have the value of 'k', you possess the complete equation governing the relationship between 'x' and 'y'. The final step involves using this established equation to find an unknown value of 'y' when a new value of 'x' is provided, or vice versa. This is achieved by substituting the known variable and the calculated 'k' into the equation and solving for the unknown. Always remember to simplify any fractions you obtain, as this leads to a cleaner and more understandable answer. Mastering these steps provides a clear roadmap for solving any direct variation problem you encounter, transforming potentially daunting equations into manageable calculations. The process is systematic and repeatable, ensuring accuracy and confidence in your mathematical endeavors.

In conclusion, direct variation describes a fundamental proportional relationship between two variables, mathematically represented by y = kx. By using a given set of values to find the constant of variation 'k', we unlock the ability to predict unknown values for either variable. In our specific problem, we found that y = (1/12)x. This enabled us to determine that when x is 8, y is 2/3. This principle extends far beyond textbook examples, appearing in countless real-world scenarios.

For further exploration into the fascinating world of mathematical relationships and proportions, you can visit **

Khan Academy** or consult resources from **

Math is Fun**.