Direct variation

Direct variation is a mathematical relationship between two variables where an increase in one results in a proportional increase in the other; this relationship is expressed as \\( y = kx \\), where \\( k \\) is the constant of variation. To identify direct variation, check if the ratio \\( \\frac{y}{x} \\) remains consistent for different values. Remember, in direct variation, when \\( x \\) doubles, \\( y \\) also doubles, highlighting a constant ratio throughout.

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    Direct Variation

    Direct variation is a relationship between two variables in which one variable is a constant multiple of the other. This means that as one variable increases or decreases, the other does so proportionally.

    Definition

    In a direct variation, the relationship is represented by the equation \[ y = kx \] where k is a non-zero constant called the constant of variation.

    Example

    If you have the direct variation equation \( y = 5x \), and you know that \( x = 2 \), you can find \( y \) by substituting \( x \) into the equation: \[ y = 5(2) = 10 \]

    Direct variation is often seen in real-world scenarios. For instance, the relationship between the distance travelled by a car and time, assuming a constant speed, is a direct variation. Here, distance is directly proportional to time, with the constant of variation being the speed of the car.

    In direct variation, if the constant of variation k is positive, both variables increase together. If k is negative, one variable increases while the other decreases.

    Direct variation is also known as a linear relationship without an intercept. This means that the graph of a direct variation equation is a straight line passing through the origin.To determine if a set of data points follows a direct variation, plot the points on a graph. If they form a straight line through the origin, you have direct variation. Another method involves checking the ratio \( \frac{y}{x} \) for all data points. If the ratio (constant k) remains the same, the data represents direct variation.

    Direct Variation Formula

    In mathematics, a direct variation describes a specific linearly dependent relationship between two variables. This relationship is defined by a formula that expresses one variable as a multiple of the other.

    Definition

    A direct variation between two variables can be expressed with the equation \( y = kx \), where k is the constant of variation. This constant k must be non-zero.

    Example

    Consider an example where the relationship between two variables is given by the equation \( y = 3x \). If \( x = 4 \), you can find \( y \) by substituting the value of \( x \): \[ y = 3(4) = 12 \] Hence, when \( x = 4 \), \( y = 12 \) in this direct variation.

    Direct variation is widely applicable in everyday scenarios. One common example is the relationship between the amount of fuel consumed by a car and the distance travelled. If the car's fuel consumption rate is constant, the amount of fuel (\( y \)) consumed is directly proportional to the distance (\( x \)) travelled. Here, the constant of variation is the fuel consumption rate.

    In a direct variation, the graph of the equation \( y = kx \) is a straight line passing through the origin (0,0).

    To determine if a set of data follows a direct variation, you can use the following methods:

    • Plot the data points on a graph. If the points form a straight line that passes through the origin, then the data represents a direct variation.
    • Calculate the ratio \( \frac{y}{x} \) for each data point. If the ratio is constant across all data points, the data is indicative of direct variation.
    In mathematics, the concept of direct variation is central to understanding linear relationships without any constant intercept term. The importance lies in its simplicity and direct proportionality, enabling straightforward calculations and interpretations.

    Direct Variation Equation

    Direct variation refers to a relationship between two variables whereby one variable is a constant multiple of the other. This concept can be expressed algebraically and is useful in understanding linear relationships without any intercept.

    Definition

    In a direct variation, the equation is given by \( y = kx \) where \( k \) is a non-zero constant termed as the constant of variation.

    Properties of the Direct Variation Equation

    Here are some key properties of the direct variation equation:

    • The relationship between \( y \) and \( x \) is linear.
    • The graph of the equation \( y = kx \) is a straight line that passes through the origin (0,0).
    • If \( k \) is positive, both variables increase together. If \( k \) is negative, one variable increases while the other decreases.

    The constant of variation \( k \) can be found by dividing \( y \) by \( x \): \( k = \frac{y}{x} \).

    Example

    Let's consider an example to understand the direct variation equation better. Suppose you have the equation \( y = 4x \). If \( x = 3 \), substitute \( x \) into the equation to find \( y \): \[ y = 4(3) = 12 \] Hence, when \( x = 3 \), \( y = 12 \) in this direct variation.

    Real-World Application

    Direct variation can be observed in numerous real-world scenarios. For instance, consider the relationship between the distance travelled by a car and the amount of time driven, assuming a constant speed. In this case, the distance \( y \) is directly proportional to the time \( x \), with the constant of variation being the speed of the car.

    To verify if a set of data represents direct variation, you can:

    • Plot the data points on a graph. If the points form a straight line that passes through the origin, then the data represents a direct variation.
    • Calculate the ratio \( \frac{y}{x} \) for each data point. If the ratio remains constant for all data points, it indicates direct variation.
    This concept is essential in understanding linear relationships without any intercept term.

    The direct variation formula is a fundamental concept in algebra that provides a foundation for understanding more complex mathematical relationships. By recognising the simple proportional relationship between two variables, you can make accurate predictions and understand the behaviour of various systems.

    Direct Variation Examples

    Direct variation in mathematics can be best understood through practical examples. These examples help clarify the relationship between variables. Understanding direct variation through examples is essential for grasping the concept fully.

    Simple Direct Variation Examples

    Consider the equation \( y = 3x \). Suppose \( x = 2 \), you can find \( y \) by substituting \( x \) into the equation: \( y = 3(2) = 6 \) Hence, when \( x = 2 \), \( y = 6 \).

    Let's look at another simple case: \( y = -4x \). If \( x = 5 \), then: \( y = -4(5) = -20 \) In this case, when \( x = 5 \), \( y = -20 \).

    A deeper dive into the concept reveals that direct variation equations always produce straight-line graphs passing through the origin. This holds true regardless of the constant of variation's value. Additionally, you can check if different sets of \( x \) and \( y \) values represent direct variation by calculating their ratios. If all ratios are equal, the relationship is a direct variation.

    Real-Life Direct Variation Examples

    Direct variation often occurs in real-life scenarios, making it a highly practical concept.

    Consider the relationship between the number of hours worked and the wages earned. If you earn $10 per hour, the total wages \( y \) can be expressed as \( y = 10x \), where \( x \) is the number of hours worked.If you work for 8 hours, the total wages would be:\( y = 10(8) = 80 \)$

    Another common example is the relationship between distance travelled and time when moving at a constant speed. If you drive at a speed of 60 km/h, the distance \( y \) travelled can be represented as \( y = 60x \), where \( x \) is the time in hours.

    Always remember, in direct variation, if \( k \) is positive, both variables increase simultaneously. If \( k \) is negative, one variable increases while the other decreases.

    Direct Variation Problems

    Solving direct variation problems requires you to establish the relationship between variables and then find the constant of variation. Typically, you'll be given some values and asked to find others.

    For instance, if \( y \text{ varies directly with } x \) and \( y = 24 \) when \( x = 8 \), determine the equation of the direct variation:Using the given values, find the constant of variation \( k \):\( k = \frac{y}{x} = \frac{24}{8} = 3 \)Thus, the equation is \( y = 3x \).

    You might encounter inverse problems where you are given the equation and asked to find specific variables. For example, using \( y = 3x \), find \( y \) when \( x = 7 \).

    Solving Direct Variation Problems

    Step-by-step approaches to solving direct variation problems can help you better understand the process. Here is a systematic method:

    • Identify the variables and determine if the relationship is direct variation.
    • Find the constant of variation \( k \).
    • Use the equation \( y = kx \) to solve for the required variable.

    Consider a more complex problem where you combine various direct variations across multiple scenarios. For example, if the distance \( y \) travelled by a car varies directly with time \( x \), and the speed \( z \) varies directly with fuel consumption \( w \), you may need to calculate \( y \) given values for \( z \) and \( w \). This involves using multiple direct variation equations simultaneously and finding relationships between multiple constants of variation.

    Direct variation - Key takeaways

    • Direct variation: A relationship where one variable is a constant multiple of another.
    • Direct variation equation: Represented by the formula y = kx, where k is the constant of variation.
    • Examples: Real-world examples include the relationship between distance travelled by a car and time, and the relationship between hours worked and wages earned.
    • Graphical representation: The graph of a direct variation formula is a straight line passing through the origin (0,0).
    • Determining direct variation: Can be identified by plotting data points on a graph or ensuring the ratio y/x remains constant.
    Frequently Asked Questions about Direct variation
    What is the formula for direct variation?
    The formula for direct variation is \\( y = kx \\), where \\( y \\) and \\( x \\) vary directly, and \\( k \\) is the constant of variation.
    What are some real-world examples of direct variation?
    Real-world examples of direct variation include the relationship between distance travelled and time at a constant speed, the cost of items and the number of items purchased when the price per item is constant, and the weight of an object and its mass under consistent gravitational force.
    How do you identify direct variation from a table of values?
    To identify direct variation from a table of values, check if the ratio of corresponding values of \\( y \\) and \\( x \\) (\\( \\frac{y}{x} \\)) is constant for all pairs. If this ratio remains the same throughout the table, the relationship represents direct variation.
    Can direct variation have a negative constant of proportionality?
    Yes, direct variation can have a negative constant of proportionality. This would mean as one variable increases, the other variable decreases proportionally. For example, if y varies directly with x and the constant is -3, then y = -3x.
    How does direct variation differ from inverse variation?
    In direct variation, as one variable increases, the other also increases proportionally, represented by \\( y = kx \\). In inverse variation, as one variable increases, the other decreases proportionally, represented by \\( y = \\frac{k}{x} \\).

    Test your knowledge with multiple choice flashcards

    If the number of hours worked and wages earned vary directly, and you earn \( £10 \) per hour, what are your total wages for working 8 hours?

    What is the equation that represents a direct variation?

    What is the general formula for direct variation?

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