Direct Variation

Definition

Direct variation is a relationship between two variables where one variable is proportional to the other. As one variable increases, the other variable increases by the same proportional amount.

5 Must Know Facts For Your Next Test

1. Direct variation is characterized by a linear relationship between two variables, where the variables increase or decrease at the same rate.
2. The relationship between the two variables in direct variation can be expressed as $y = kx$, where $k$ is the constant of proportionality.
3. The constant of proportionality, $k$, represents the rate of change between the two variables and is a fixed ratio that describes their proportional relationship.
4. Direct variation is often used to model real-world situations, such as the relationship between the cost of an item and the quantity purchased, or the relationship between the distance traveled and the time taken.
5. Solving for a specific variable in a formula that involves direct variation requires isolating the variable and using the constant of proportionality to determine the value.

Review Questions

• Explain how direct variation can be used to solve for a specific variable in a formula.
  • When a formula involves direct variation, the relationship between the variables can be expressed as $y = kx$, where $k$ is the constant of proportionality. To solve for a specific variable, you would need to isolate that variable by rearranging the formula and using the known constant of proportionality. For example, if the formula is $V = \frac{4}{3}\pi r^3$, where $V$ represents volume and $r$ represents radius, you could solve for the radius by rearranging the formula to $r = \sqrt[3]{\frac{3V}{4\pi}}$, using the direct variation relationship between volume and radius.
• Describe how direct variation can be used to solve applications involving rational equations.
  • In applications involving rational equations, direct variation can be used to model the relationship between the variables. For example, in the equation $\frac{x}{y} = k$, where $k$ is the constant of proportionality, $x$ and $y$ are directly varying quantities. To solve such applications, you would need to isolate the variable of interest by manipulating the rational equation and using the known constant of proportionality. This allows you to determine the value of the variable based on the given information in the problem statement.
• Analyze the role of the constant of proportionality in understanding and applying direct variation to solve problems.
  • The constant of proportionality, $k$, is a crucial element in understanding and applying direct variation. It represents the fixed ratio that describes the proportional relationship between the two variables. By identifying the constant of proportionality, you can determine the rate of change between the variables and use this information to solve for a specific variable in a formula or application involving direct variation. The constant of proportionality allows you to establish the underlying linear relationship between the variables and use it to make informed calculations and predictions, which is essential for solving problems in the context of 2.3 Solve a Formula for a Specific Variable and 7.5 Solve Applications with Rational Equations.

Related terms

Proportionality: A relationship between two variables where one variable is a constant multiple of the other variable.

Constant of Proportionality: The fixed ratio that describes the proportional relationship between two directly varying quantities.

Linear Equation: An equation where the variables are raised to the first power and the variables are connected by addition, subtraction, or equality.