Direct and inverse variation are fundamental concepts in algebra that describe relationships between variables. These concepts help us understand how changes in one quantity affect another, whether they increase together or move in opposite directions.

In this section, we'll explore how to identify, solve, and apply direct and inverse variation problems. We'll also compare these relationships, examine real-world examples, and learn how to represent them graphically. Understanding these concepts is crucial for solving many practical problems in science, economics, and everyday life.

Direct and Inverse Variation

Direct variation problem solving

• Direct variation occurs when two quantities $x$ and $y$ are related by the equation $y = kx$, where $k$ is a non-zero constant called the constant of variation
• Identify direct variation by observing if $y$ increases as $x$ increases, or if $y$ decreases as $x$ decreases
• Solve for the constant of variation ($k$) by substituting two corresponding values of $x$ and $y$ into the equation $y = kx$ and solving for $k$
• Find the value of one quantity given the other by substituting the known value of $x$ or $y$ and the constant of variation ($k$) into the equation $y = kx$ and solving for the unknown quantity
• Real-world example: The cost of apples ($y$) is directly proportional to the number of pounds purchased ($x$)
  • If 3 pounds of apples cost $6, find the cost of 5 pounds of apples
• The equation $y = kx$ represents a linear function, where $x$ is the independent variable and $y$ is the dependent variable

Applications of inverse variation

• Inverse variation occurs when two quantities $x$ and $y$ are related by the equation $y = \frac{k}{x}$, where $k$ is a non-zero constant
• Identify inverse variation by observing if $y$ decreases as $x$ increases, or if $y$ increases as $x$ decreases
• Solve for the constant of variation ($k$) by substituting two corresponding values of $x$ and $y$ into the equation $y = \frac{k}{x}$ and solving for $k$
• Apply inverse variation to real-world problems such as:
  • The time it takes to fill a pool is inversely proportional to the number of hoses used
    • If it takes 6 hours to fill a pool with 2 hoses, how long will it take to fill the same pool with 3 hoses?
  • The intensity of light is inversely proportional to the square of the distance from the light source
    • If the intensity of light is 100 lux at 2 meters from the source, find the intensity at 4 meters from the source

Direct vs inverse relationships

• Recognize direct variation in real-world scenarios like:
  • The distance traveled by a car is directly proportional to the time spent driving at a constant speed
    • If a car travels 120 miles in 2 hours, find the distance traveled in 3 hours
  • The amount of money earned is directly proportional to the number of hours worked at a constant hourly rate
    • If a person earns $75 for working 5 hours, find the amount earned for working 8 hours
• Recognize inverse variation in real-world scenarios like:
  • The number of days it takes to complete a project is inversely proportional to the number of workers assigned to the project
    • If 6 workers can complete a project in 10 days, how many days will it take 4 workers to complete the same project?
  • The pressure of a gas is inversely proportional to its volume at a constant temperature (Boyle's Law)
    • If the pressure of a gas is 2 atm when its volume is 5 liters, find the pressure when the volume is decreased to 2 liters
• Compare direct and inverse variation by noting that in direct variation, the ratio $\frac{y}{x}$ is constant ($y = kx$), while in inverse variation, the product $xy$ is constant ($y = \frac{k}{x}$)

Graphical representation and analysis

• The graph of a direct variation is a straight line passing through the origin (0,0)
• The graph of an inverse variation is a hyperbola
• Use graphs to visualize the relationship between variables in direct and inverse variation
• Analyze proportions in both direct and inverse variation equations to solve for unknown values