Z Varies Directly With X And Inversely With Y

Z Varies Directly with X and Inversely with Y: A Comprehensive Guide

Understanding relationships between variables is fundamental in mathematics and numerous scientific fields. One such relationship, often encountered in physics, engineering, and economics, is where one variable varies directly with another and inversely with a third. This article delves deep into the concept of "z varies directly with x and inversely with y," exploring its meaning, representation, applications, and problem-solving techniques. We'll cover the core concepts, provide worked examples, and offer strategies for mastering this essential mathematical principle.

Understanding Direct and Inverse Variation

Before diving into the combined variation, let's solidify our understanding of direct and inverse variations individually.

Direct Variation

In a direct variation, two variables are proportionally related. This means that as one variable increases, the other increases proportionally, and vice versa. The relationship can be expressed as:

z = kx

where:

• 'z' and 'x' are the variables.
• 'k' is a constant of proportionality. This constant represents the rate of change between the two variables.

Example: If the cost (z) of apples is directly proportional to the number of apples (x) purchased, and the cost of one apple is $2 (k), then the relationship is z = 2x. If you buy 5 apples, the cost will be z = 2 * 5 = $10.

Inverse Variation

In an inverse variation, as one variable increases, the other decreases proportionally, and vice-versa. The relationship is represented as:

z = k/x

where:

• 'z' and 'x' are the variables.
• 'k' is the constant of proportionality.

Example: The time (z) it takes to travel a fixed distance is inversely proportional to the speed (x). If the distance is 100 miles and the speed is 50 mph, the time taken is z = 100/50 = 2 hours. If the speed increases to 100 mph, the time decreases to z = 100/100 = 1 hour.

Z Varies Directly with X and Inversely with Y: The Combined Variation

Now, let's combine these concepts. When we say "z varies directly with x and inversely with y," it means that z is directly proportional to x and inversely proportional to y. This relationship is expressed mathematically as:

z = kx/y

where:

• 'z', 'x', and 'y' are the variables.
• 'k' is the constant of proportionality.

This equation signifies that:

• If x increases, z increases proportionally (direct variation with x).
• If y increases, z decreases proportionally (inverse variation with y).

Determining the Constant of Proportionality (k)

To fully define the relationship, we need to determine the value of 'k'. This is done by using a known set of values for z, x, and y.

Example: Suppose z varies directly with x and inversely with y. If z = 6 when x = 3 and y = 2, find the constant of proportionality (k) and the equation relating z, x, and y.

1. Substitute the known values into the equation: 6 = k * (3/2)
2. Solve for k: k = (6 * 2) / 3 = 4

Therefore, the constant of proportionality is 4, and the equation relating z, x, and y is:

z = 4x/y

Solving Problems Involving Combined Variation

Let's explore several problem-solving scenarios involving this type of variation.

Example 1: Finding a Missing Variable

The volume (z) of a gas varies directly with its temperature (x) and inversely with its pressure (y). If the volume is 10 liters when the temperature is 200 Kelvin and the pressure is 5 atmospheres, what is the volume when the temperature is 300 Kelvin and the pressure is 10 atmospheres?

1. Find k: 10 = k * (200/5) => k = 1/4
2. Write the equation: z = (1/4)x/y
3. Substitute new values: z = (1/4) * (300/10) = 7.5 liters

Therefore, the volume is 7.5 liters under the new conditions.

Example 2: Word Problem Application

The number of days (z) it takes to complete a project varies directly with the amount of work (x) and inversely with the number of workers (y). If it takes 10 days to complete a project with 20 units of work and 4 workers, how many days will it take to complete a project with 30 units of work and 6 workers?

1. Find k: 10 = k * (20/4) => k = 2
2. Write the equation: z = 2x/y
3. Substitute new values: z = 2 * (30/6) = 10 days

Therefore, it will still take 10 days to complete the project with the adjusted work and workers.

Real-World Applications

The concept of "z varies directly with x and inversely with y" has numerous applications across various disciplines:

• Physics: Gas laws (Boyle's Law, Charles's Law, and the Combined Gas Law) illustrate this relationship beautifully. The volume of a gas is directly proportional to its temperature and inversely proportional to its pressure.

• Engineering: Stress and strain in materials exhibit similar relationships. Stress is directly proportional to force and inversely proportional to the cross-sectional area.

• Economics: The price of a commodity can be directly related to demand and inversely related to supply.

• Chemistry: Reaction rates often vary directly with reactant concentrations and inversely with reaction time.

Advanced Concepts and Extensions

While the basic equation z = kx/y provides a strong foundation, more complex variations exist, involving more than three variables or incorporating powers and roots. For instance, z could vary directly with the square of x and inversely with the cube root of y, represented as:

z = kx²/∛y

Mastering these variations requires a solid understanding of algebraic manipulation and problem-solving skills.

Conclusion: Mastering Combined Variation

Understanding combined variations, specifically where 'z varies directly with x and inversely with y,' is a crucial skill across multiple scientific and mathematical disciplines. By grasping the fundamental concepts, applying the appropriate equations, and practicing problem-solving, you can confidently tackle a wide array of challenges involving this type of relationship. Remember to break down complex problems into manageable steps, carefully substitute values, and meticulously solve for the unknowns. With dedicated effort and practice, you can master this essential mathematical concept and apply it to various real-world scenarios. The key is consistent practice and a solid understanding of the underlying principles of direct and inverse variation. Continue exploring more complex variations and their applications to further solidify your understanding and expand your problem-solving capabilities.