Solving Direct Variation Problems Finding Equations And Values

Hey guys! Today, we're diving into the fascinating world of direct variation, a concept that pops up frequently in mathematics and real-life scenarios. We'll tackle a specific problem where y varies directly as x², and by the end of this guide, you'll be a pro at solving these types of problems. So, buckle up, and let's get started!

Problem Overview

Our mission, should we choose to accept it (and you totally should!), is to solve a problem where we're told that y varies directly as x². We're given a specific data point: y = -12 when x = 2. Using this information, we need to:

(a) Find the equation that relates x and y. (b) Determine the values of x when y = -27.

Sounds a bit intimidating? Don't worry! We'll break it down step-by-step, making it super easy to understand.

(a) Finding the Equation Relating x and y

Understanding Direct Variation

First things first, let's clarify what it means when we say "y varies directly as x²." In mathematical terms, this means that y is directly proportional to the square of x. We can express this relationship as an equation:

y = k * x²

Where:

• y is our dependent variable.
• x is our independent variable.
• k is the constant of variation, a crucial value that determines the specific relationship between y and x².

Our goal here is to find the value of k, which will give us the exact equation linking x and y.

Using the Given Data

We're given a valuable piece of information: y = -12 when x = 2. This is our golden ticket to finding k! We can simply substitute these values into our direct variation equation:

-12 = k * (2)²

Solving for k

Now, let's solve for k. First, we simplify the equation:

-12 = k * 4

To isolate k, we divide both sides of the equation by 4:

k = -12 / 4 k = -3

So, we've found our constant of variation! k = -3. This tells us the strength and direction of the relationship between y and x².

The Equation Relating x and y

Now that we know k, we can write the complete equation that relates x and y. We simply substitute k = -3 back into our direct variation equation:

y = -3x²

Voila! We've found the equation. This equation tells us exactly how y changes in relation to x². For every value of x, we can square it, multiply by -3, and find the corresponding value of y. Isn't that neat?

(b) Finding the Values of x when y = -27

Using the Equation

Now that we have our equation, y = -3x², we can use it to solve the second part of the problem: finding the values of x when y = -27. This is where our algebraic skills come into play!

We simply substitute y = -27 into our equation:

-27 = -3x²

Solving for x

Our next step is to isolate x². We can do this by dividing both sides of the equation by -3:

x² = -27 / -3 x² = 9

Now, we have x² = 9. To find x, we need to take the square root of both sides of the equation. Remember, when we take the square root, we get two possible solutions: a positive and a negative value.

x = ±√9 x = ±3

The Values of x

So, we have two values for x: x = 3 and x = -3. This means that when y = -27, x can be either 3 or -3. Both of these values satisfy our direct variation equation.

Putting It All Together

Let's recap what we've done. We started with the information that y varies directly as x² and that y = -12 when x = 2. We then:

1. Found the constant of variation, k, by substituting the given values into the direct variation equation.
2. Wrote the equation relating x and y: y = -3x².
3. Found the values of x when y = -27 by substituting y = -27 into the equation and solving for x.

We discovered that x = 3 and x = -3 when y = -27.

Key Takeaways

Here are some key takeaways from this problem:

• Direct Variation: When y varies directly as x², it means y = k * x², where k is the constant of variation.
• Finding the Constant of Variation: Use given data points to substitute into the direct variation equation and solve for k.
• Solving for Variables: Once you have the equation, you can substitute values for one variable and solve for the other.
• Square Roots: Remember to consider both positive and negative solutions when taking the square root.

Common Mistakes to Avoid

To ensure you ace these problems, here are some common mistakes to watch out for:

• Forgetting the Constant of Variation: Always include the constant of variation, k, in your direct variation equation.
• Incorrectly Squaring x: Make sure you're squaring x correctly when dealing with x².
• Ignoring Negative Solutions: Don't forget to consider both positive and negative solutions when taking the square root.
• Misunderstanding Direct Variation: Make sure you understand the concept of direct variation and how it translates into an equation.

Real-World Applications

Direct variation isn't just a mathematical concept; it pops up in various real-world situations. For example:

• The area of a circle varies directly as the square of its radius. This means that if you double the radius of a circle, its area increases by a factor of four.
• The distance a freely falling object travels varies directly as the square of the time it falls. This explains why objects fall faster and faster as they descend.
• The electrical power dissipated in a resistor varies directly as the square of the current flowing through it. This is a fundamental principle in electrical engineering.

Understanding direct variation helps us model and predict how different quantities relate to each other in these real-world scenarios.

Practice Makes Perfect

The best way to master direct variation is to practice! Try solving similar problems with different values and scenarios. Challenge yourself to identify real-world examples of direct variation and express them mathematically.

Conclusion

So there you have it! We've successfully navigated a problem involving direct variation, found the equation relating x and y, and determined the values of x for a given y. Remember, the key is to understand the concept of direct variation, use the given information wisely, and apply your algebraic skills. Keep practicing, and you'll become a direct variation whiz in no time!

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