Combined Variation - Algebra-2

1. Fundamental Concepts

Definition: Combined variation occurs when a quantity varies directly with the product of two or more other quantities and inversely with one or more other quantities.
Direct Variation: A relationship where one quantity increases as the other increases, expressed as $$y = kx$$.
Inverse Variation: A relationship where one quantity decreases as the other increases, expressed as $$y = \frac{k}{x}$$.

2. Key Concepts

Basic Rule: $$y = k \cdot x_1 \cdot x_2 \cdot \ldots \cdot \frac{1}{x_n}$$

Degree Preservation: The highest degree in the result matches input

Application: Used to model real-world scenarios such as physics and engineering problems

3. Examples

Example 1 (Basic)

Problem: If $$y$$ varies directly as $$x$$ and inversely as $$z$$, and $$y = 6$$ when $$x = 2$$ and $$z = 3$$, find $$y$$ when $$x = 4$$ and $$z = 6$$.

Step-by-Step Solution:

Given: $$y = k \cdot \frac{x}{z}$$
Substitute known values: $$6 = k \cdot \frac{2}{3}$$
Solve for $$k$$: $$k = 9$$
Find $$y$$ when $$x = 4$$ and $$z = 6$$: $$y = 9 \cdot \frac{4}{6} = 6$$

Validation: Substitute $$x=4$$ and $$z=6$$ → Original: $$y = 6$$; Simplified: $$y = 6$$ ✓

Example 2 (Intermediate)

Problem: If $$y$$ varies jointly as $$x$$ and $$z$$ and inversely as $$w$$, and $$y = 10$$ when $$x = 2$$, $$z = 5$$, and $$w = 2$$, find $$y$$ when $$x = 4$$, $$z = 3$$, and $$w = 6$$.

Step-by-Step Solution:

Given: $$y = k \cdot \frac{x \cdot z}{w}$$
Substitute known values: $$10 = k \cdot \frac{2 \cdot 5}{2}$$
Solve for $$k$$: $$k = 2$$
Find $$y$$ when $$x = 4$$, $$z = 3$$, and $$w = 6$$: $$y = 2 \cdot \frac{4 \cdot 3}{6} = 4$$

Validation: Substitute $$x=4$$, $$z=3$$, and $$w=6$$ → Original: $$y = 10$$; Simplified: $$y = 4$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use graphs to visualize direct and inverse relationships.
Error-Proofing: Always check units and dimensions to ensure consistency.
Concept Reinforcement: Practice with a variety of problems to reinforce understanding.