Determining an Inverse Variation - Algebra-2

1. Fundamental Concepts

Definition: Inverse variation describes a relationship between two variables where their product is always a constant. If $x$ and $y$ are inversely proportional, then $xy = k$, where $k$ is a constant.
Graphical Representation: The graph of an inverse variation is a hyperbola with the equation $y = \frac{k}{x}$.
Real-World Applications: Inverse variation can be observed in various contexts such as physics (e.g., Boyle's Law for gases), economics (demand and supply), and more.

2. Key Concepts

Basic Rule: $$xy = k$$

Degree Preservation: The product of $x$ and $y$ remains constant regardless of changes in $x$ or $y$.

Application: Used to model relationships where one variable increases while the other decreases proportionally.

3. Examples

Example 1 (Basic)

Problem: If $y$ varies inversely with $x$ and $y = 6$ when $x = 2$, find the value of $y$ when $x = 3$.

Step-by-Step Solution:

Identify the constant $k$: Since $xy = k$, we have $6 \cdot 2 = k$, so $k = 12$.
Find $y$ when $x = 3$: Using $xy = 12$, we get $3y = 12$, so $y = 4$.

Validation: Substitute $x = 3$ and $y = 4$ into the original equation $xy = 12$. $3 \cdot 4 = 12$ ✓

Example 2 (Intermediate)

Problem: The pressure $P$ of a gas varies inversely with its volume $V$. If the pressure is 500 Pa when the volume is 2 liters, find the pressure when the volume is increased to 5 liters.

Step-by-Step Solution:

Identify the constant $k$: Since $PV = k$, we have $500 \cdot 2 = k$, so $k = 1000$.
Find $P$ when $V = 5$: Using $PV = 1000$, we get $5P = 1000$, so $P = 200$ Pa.

Validation: Substitute $V = 5$ and $P = 200$ into the original equation $PV = 1000$. $5 \cdot 200 = 1000$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use graphs to visualize the inverse relationship between variables.
Error-Proofing: Always check that the product of the variables equals the constant $k$.
Concept Reinforcement: Practice with real-world examples to understand the practical implications of inverse variation.