Problem 13 - Algebra 2

Question

If \(y\) varies inversely as \(x\) and \(y=16\) when \(x=5,\) find \(y\) when \(x=20\)

Step-by-Step Solution

Answer

When \( x = 20 \), \( y = 4. \)

Step 1: Understanding Inverse Variation

Inverse variation means that when one variable increases, the other decreases in such a way that the product remains constant. We express this relationship with the formula: \[ y = \frac{k}{x} \] where \( k \) is the constant of variation.

Step 2: Finding the Constant of Variation

Given that \( y = 16 \) when \( x = 5 \), we can plug these values into the inverse variation formula to find \( k \):\[ 16 = \frac{k}{5} \]Multiplying both sides by 5 to solve for \( k \), we get:\[ k = 16 \times 5 = 80 \]

Step 3: Using the Constant to Find the New Value

Now that we know the constant \( k = 80 \), we use it to find \( y \) when \( x = 20 \). Plug \( x = 20 \) into the inverse variation formula:\[ y = \frac{80}{20} \] Divide to find \( y \):\[ y = 4 \]

Key Concepts

Constant of Variation

In the realm of inverse variation, the constant of variation is a pivotal element. It represents a fixed value that helps maintain the balance between two inversely related variables. When one variable increases, the other decreases, but their product remains steady due to this constant. Consider the equation \( y = \frac{k}{x} \). Here, \( k \) symbolizes the constant of variation.
This constant is crucial because it is the anchor that makes mathematical equations reliable and predictable. In our example, when \( y = 16 \) and \( x = 5 \), we can determine the constant \( k \) by rearranging and substituting these into the formula:

• Plug in the known values to get \( 16 = \frac{k}{5} \).
• Multiply both sides by \( 5 \) to isolate \( k \), resulting in \( k = 16 \times 5 = 80 \).

Once the constant is found, it remains unchanged throughout the problem, signifying that no matter how \( x \) or \( y \) vary, their product equals \( k \). Understanding this ensures you can solve similar problems effectively.

Inverse Relationship

An inverse relationship is a type of connection between two variables where they move in opposite directions. As one increases, the other decreases, creating a unique balancing dynamic. This relationship is mathematically expressed through the formula \( y = \frac{k}{x} \), indicating that \( y \) varies inversely as \( x \).
This concept can be understood best by visualizing a seesaw. When one side goes up, the other must go down. The key to this balancing act is the constant \( k \), which ensures their product remains steady, something we've demonstrated in the step-by-step solution.

• Start with the equation \( y = \frac{k}{x} \).
• To see if variables have an inverse relationship, simply check if their product is equal across different scenarios.
• In our case, knowing \( y = 16 \) and \( x = 5 \) computed \( k = 80 \), allowing us to find new \( y \) values for different \( x \).

This principle of inverse relationship finds applications in various fields, ranging from physics to economics, demonstrating its wide-ranging significance.

Algebraic Problem-Solving

Algebraic problem-solving with inverse variation involves harnessing mathematical tools and logical reasoning to decode the relationships between variables. The given problem is a great example of how algebra can be wielded to find unknown quantities.
To solve an inverse variation problem, follow a step-by-step approach:

• Identify the type of variation or relationship given in the problem. Here, it's an inverse relationship.
• Determine the known values and set up the general formula \( y = \frac{k}{x} \).
• Solve for the constant of variation \( k \) using provided values, doing so by substituting and rearranging the equation.
• Use the constant of variation to find unknown values across different scenarios.

In our sample problem, knowing \( y = 16 \) when \( x = 5 \) allowed us to find \( k = 80 \). Applying the constant to a new condition where \( x = 20 \) enabled us to calculate \( y = 4 \).
By mastering these algebraic steps, students can efficiently solve inverse variation problems and apply these skills to a variety of mathematical challenges.