Solving Inverse Variation Problems Finding Z When X Is 4 And Y Is 9

In the realm of mathematical relationships, inverse variation plays a crucial role in describing how quantities interact. When one quantity increases, the other decreases proportionally, creating a fascinating dance of interdependence. This article delves into the concept of inverse variation, using a specific example to illustrate how to determine the value of a variable when others change. Let's explore the intricacies of this relationship and uncover the solution to our problem.

Understanding Inverse Variation

At its core, inverse variation describes a relationship where two variables move in opposite directions. Mathematically, we express this as z varies inversely with the product of x and y. This means that as the product of x and y increases, z decreases, and vice versa. The fundamental equation that governs this relationship is:

z = k / (x * y)

Where:

• z is the dependent variable
• x and y are independent variables
• k is the constant of variation

The constant of variation, k, is the linchpin of this relationship. It's a fixed value that dictates the strength of the inverse variation. To find k, we need a set of values for x, y, and z. Once we have k, we can use it to determine z for any given pair of x and y values.

Identifying Inverse Variation in Real-World Scenarios: You might be wondering, where does inverse variation show up in the real world? The answer is, quite frequently! Consider these examples:

• Speed and Time: The time it takes to travel a certain distance varies inversely with the speed. If you increase your speed, the travel time decreases, and vice-versa.
• Pressure and Volume: In a closed container, the pressure of a gas varies inversely with its volume (Boyle's Law). Squeeze the container (decrease the volume), and the pressure increases.
• Workers and Time: The time it takes to complete a task varies inversely with the number of workers. More workers mean less time to finish the job.

These are just a few examples, and inverse variation pops up in physics, engineering, economics, and many other fields. Understanding this relationship helps us predict how changes in one variable will impact others.

Problem Setup: Deciphering the Given Information

Our problem presents us with a classic inverse variation scenario. We're told that z varies inversely with the product of x and y. To kick things off, let's restate the problem and dissect the given information:

• z varies inversely with the product of x and y.
• When x = 2 and y = 4, z = 0.5
• We need to find the value of z when x = 4 and y = 9.

Now, let's translate this information into the language of mathematics. We know that the relationship between z, x, and y can be expressed as:

z = k / (x * y)

Our first task is to determine the constant of variation, k. We can do this by plugging in the given values of x, y, and z (x = 2, y = 4, z = 0.5) into the equation:

0. 5 = k / (2 * 4)

This equation allows us to isolate k and find its value. Once we have k, we'll be equipped to tackle the second part of the problem: finding z when x = 4 and y = 9.

The Importance of Clear Problem Definition: It's tempting to jump straight into calculations, but taking a moment to clearly define the problem is crucial. This step ensures we understand the relationships between variables and what we're trying to solve. Breaking down the problem into smaller parts, like finding k first, makes the overall solution process more manageable.

Step-by-Step Solution: Unraveling the Mystery

Now, let's embark on the journey of solving this problem step by step. Our goal is to find the value of z when x = 4 and y = 9. But first, we need to determine the constant of variation, k.

Step 1: Finding the Constant of Variation (k)

We have the equation:

0. 5 = k / (2 * 4)

Simplify the denominator:

0. 5 = k / 8

To isolate k, multiply both sides of the equation by 8:

0. 5 * 8 = k

4 = k

So, the constant of variation, k, is 4. This value is the key to unlocking the relationship between z, x, and y in this specific problem.

Step 2: Using k to Find z When x = 4 and y = 9

Now that we know k = 4, we can use the general equation for inverse variation and plug in the new values of x and y (x = 4, y = 9) to find z:

z = k / (x * y)

z = 4 / (4 * 9)

Simplify the denominator:

z = 4 / 36

Reduce the fraction by dividing both numerator and denominator by their greatest common divisor, which is 4:

z = 1 / 9

Therefore, the value of z when x = 4 and y = 9 is 1/9.

Checking the Answer for Reasonableness: It's always a good idea to take a moment and think about whether our answer makes sense. We know that z varies inversely with the product of x and y. When we went from the first set of values (x = 2, y = 4) to the second set (x = 4, y = 9), the product of x and y increased significantly (from 8 to 36). So, we'd expect z to decrease, which it did (from 0.5 to 1/9). This quick check gives us confidence that our answer is in the right ballpark.

Answer and Conclusion: The Final Piece of the Puzzle

After our step-by-step journey through the realm of inverse variation, we've arrived at our destination: the value of z when x = 4 and y = 9.

The answer is z = 1/9

This problem highlights the power of understanding mathematical relationships. By recognizing the inverse variation between z and the product of x and y, we were able to set up an equation, find the constant of variation, and ultimately solve for the unknown value of z. Inverse variation, while seemingly abstract, has profound implications in various fields, from physics and engineering to economics and everyday life.

Key Takeaways:

• Inverse variation describes a relationship where one variable decreases as another increases proportionally.
• The equation z = k / (x * y) is the mathematical representation of this relationship.
• The constant of variation, k, is crucial for solving inverse variation problems.
• Clear problem definition and step-by-step solutions are essential for success.

By mastering the concepts of inverse variation, you equip yourself with a valuable tool for analyzing and understanding the world around you. So, embrace the power of mathematical relationships, and continue to explore the fascinating world of numbers and equations.