How To Write An Equation For A Direct Variation

How to Write an Equation for a Direct Variation

Direct variation describes a relationship between two variables where an increase in one variable leads to a proportional increase in the other, and a decrease in one variable leads to a proportional decrease in the other. Understanding how to write an equation representing this relationship is crucial in various fields, including mathematics, physics, engineering, and economics. This comprehensive guide will explore the intricacies of direct variation, providing a step-by-step approach to formulating its equation and illustrating the concept with numerous examples.

Understanding Direct Variation

At its core, direct variation signifies a linear relationship between two variables, x and y, where the ratio of y to x remains constant. This constant is known as the constant of proportionality, often represented by the letter k. The relationship can be expressed verbally as "y varies directly with x" or "y is directly proportional to x."

Mathematically, this relationship is defined as:

y = kx

where:

• y is the dependent variable
• x is the independent variable
• k is the constant of proportionality (k ≠ 0)

The constant k dictates the rate of change; a larger k implies a steeper slope on a graph. Crucially, when x = 0, y = 0. This is a defining characteristic of direct variation – the line representing the relationship always passes through the origin (0,0).

Steps to Write an Equation for Direct Variation

To write an equation representing a direct variation, follow these steps:

Step 1: Identify the Variables

First, determine the two variables involved in the direct variation. Carefully read the problem statement to understand which variable depends on the other. Often, one variable will be explicitly stated as being "directly proportional" or "directly varying" with the other.

For instance:

• "The distance traveled (d) varies directly with the time (t) spent traveling." Here, d is the dependent variable and t is the independent variable.
• "The cost (c) of apples is directly proportional to the number of apples (n) purchased." In this scenario, c is the dependent variable and n is the independent variable.

Step 2: Determine the Constant of Proportionality (k)

This is the most crucial step. You'll need at least one pair of corresponding values for x and y to find k. Substitute these values into the equation y = kx and solve for k.

Example:

The distance (d) traveled varies directly with the time (t) spent traveling. If a car travels 150 miles in 3 hours, find the equation representing this direct variation.

1. Identify variables: d (dependent) and t (independent).
2. Substitute values: We know d = 150 miles and t = 3 hours. Substitute these into the equation: 150 = k * 3.
3. Solve for k: Divide both sides by 3: k = 150/3 = 50.

Therefore, the constant of proportionality is 50 miles per hour (mph).

Step 3: Write the Equation

Now that we have the constant of proportionality (k), we can write the complete equation for the direct variation:

d = 50t

This equation tells us that the distance traveled is equal to 50 times the time spent traveling.

Examples of Direct Variation Equations

Let's explore a range of examples to solidify our understanding.

Example 1: Simple Linear Relationship

The cost (c) of buying oranges is directly proportional to the number of oranges (n) purchased. If 5 oranges cost $2.50, find the equation.

1. Variables: c (dependent), n (independent).
2. Substitute values: 2.50 = k * 5
3. Solve for k: k = 2.50/5 = 0.5
4. Equation: c = 0.5n

This equation indicates that each orange costs $0.50.

Example 2: More Complex Scenario

The area (A) of a circle varies directly with the square of its radius (r). If a circle with a radius of 2 cm has an area of 12.57 cm², find the equation.

Note: This example demonstrates that even with a non-linear relationship (A is proportional to r²), we can still use the direct variation concept. The equation becomes:

A = kr²

1. Variables: A (dependent), r (independent).
2. Substitute values: 12.57 = k * 2² = 4k
3. Solve for k: k = 12.57/4 ≈ 3.14 (approximating to π)
4. Equation: A ≈ 3.14r² (This is the familiar formula for the area of a circle)

Example 3: Real-World Application

The amount of money earned (M) by a worker varies directly with the number of hours (h) worked. If a worker earns $100 for 8 hours of work, find the equation.

1. Variables: M (dependent), h (independent).
2. Substitute values: 100 = k * 8
3. Solve for k: k = 100/8 = 12.5
4. Equation: M = 12.5h

This equation indicates the hourly wage is $12.50.

Differentiating Direct Variation from Other Relationships

It's crucial to distinguish direct variation from other relationships, especially inverse variation and partial variation.

• Inverse Variation: In inverse variation, as one variable increases, the other decreases proportionally. The equation is of the form y = k/x.
• Partial Variation: Partial variation involves a combination of a constant term and a term representing direct variation. The equation is of the form y = mx + c, where m is the slope and c is the y-intercept (not passing through the origin).

Understanding these differences is critical for accurately modeling real-world phenomena.

Graphical Representation of Direct Variation

Direct variation is always represented graphically as a straight line passing through the origin (0,0). The slope of this line is equal to the constant of proportionality (k). A steeper slope indicates a greater rate of change.

Solving Problems Involving Direct Variation

Once you've established the equation, you can use it to solve various problems. For instance, using the equation M = 12.5h from our previous example:

• Finding earnings for a specific number of hours: If the worker works 10 hours, their earnings will be M = 12.5 * 10 = $125.
• Finding the number of hours worked for a specific amount of earnings: If the worker earned $150, the number of hours worked will be h = 150 / 12.5 = 12 hours.

Advanced Applications of Direct Variation

Beyond the basic examples, direct variation finds application in more complex scenarios:

• Physics: Hooke's Law (F = kx), relating the force exerted on a spring to its extension, is a prime example of direct variation.
• Engineering: Scaling models in engineering often relies on direct variation principles.
• Economics: Supply and demand curves sometimes exhibit direct variation in specific market conditions.

Conclusion

Mastering the skill of writing an equation for direct variation is a foundational concept in algebra and has widespread practical applications. By carefully following the steps outlined in this guide, you can confidently tackle a variety of problems involving this crucial mathematical relationship. Remember to identify the variables, determine the constant of proportionality, write the equation, and interpret the results within the context of the problem. Practice with diverse examples and applications will solidify your understanding and enhance your problem-solving abilities. Don't be afraid to explore more complex scenarios and apply your knowledge to real-world situations. This deep understanding will prove invaluable in various academic and professional pursuits.