Solving Variation Problems Finding Z When X And Y Change

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This problem delves into the concepts of inverse and direct variation, showcasing how variables can influence each other in mathematical relationships. We are presented with a scenario where the variable z exhibits a dual behavior: it varies inversely with x and directly with y. This means that as x increases, z decreases proportionally, while as y increases, z increases proportionally. To solve this, we will first establish a mathematical equation that captures this relationship, using the given initial values to determine the constant of variation. Once we have this equation, we can then use it to find the value of z under the new conditions of x and y.

Defining Direct and Inverse Variation

Before diving into the specific problem, let's clarify the core concepts of direct and inverse variation. Direct variation occurs when two variables increase or decrease together at a constant rate. Mathematically, this is expressed as y = kx, where k is the constant of variation. This means that y is directly proportional to x. A classic example of direct variation is the relationship between the number of hours worked and the amount earned, assuming a fixed hourly wage.

Conversely, inverse variation describes a situation where one variable increases as the other decreases, and vice versa. The relationship is expressed as y = k/x, where k is again the constant of variation. In this case, y is inversely proportional to x. An example of inverse variation is the relationship between the speed of a vehicle and the time it takes to travel a certain distance; as speed increases, travel time decreases.

In the problem at hand, z varies both directly and inversely with different variables. This is known as joint variation, which combines both direct and inverse relationships into a single equation. Understanding these fundamental concepts is crucial for setting up and solving the problem accurately.

Setting Up the Equation for Joint Variation

Given that z varies inversely with x and directly with y, we can express this relationship mathematically as:

z = k(y/x)

Where k is the constant of variation. This equation encapsulates the combined effect of both inverse and direct variation on z. The direct variation with y is represented by the y in the numerator, indicating that z increases proportionally with y. The inverse variation with x is represented by the x in the denominator, indicating that z decreases proportionally with x. The constant k scales this relationship and ensures that the equation holds true for all corresponding values of x, y, and z.

To find the specific value of k for this problem, we use the given initial conditions: when x = 6 and y = 2, z = 5. Substituting these values into the equation, we get:

5 = k(2/6)

This equation allows us to solve for k, which is essential for fully defining the relationship between z, x, and y. Once we have the value of k, we can use the equation to predict the value of z for any other given pair of x and y values. The constant of variation, k, essentially acts as a scaling factor that calibrates the relationship between the variables, ensuring that the equation accurately reflects the observed variations.

Solving for the Constant of Variation (k)

To determine the value of the constant of variation, k, we start with the equation derived from the problem statement and the given initial conditions:

5 = k(2/6)

Our goal is to isolate k on one side of the equation. To do this, we can first simplify the fraction 2/6 to 1/3:

5 = k(1/3)

Next, we multiply both sides of the equation by 3 to eliminate the fraction:

5 * 3 = k(1/3) * 3

This simplifies to:

15 = k

Therefore, the constant of variation, k, is 15. This value is crucial as it defines the specific relationship between z, x, and y in this particular problem. With k now known, we can rewrite the general equation for the joint variation as:

z = 15(y/x)

This equation provides a complete mathematical description of how z changes in response to changes in x and y. The constant k = 15 ensures that the equation accurately predicts the value of z for any given pair of x and y values, making it a powerful tool for solving the problem and understanding the underlying relationships between the variables.

Calculating z with New Values of x and y

Now that we have determined the constant of variation, k = 15, we can use the equation z = 15(y/x) to find the value of z when x = 4 and y = 9. This step involves substituting these new values of x and y into the equation and then solving for z. The equation serves as a mathematical model that captures the relationship between the variables, allowing us to predict the value of z under different conditions.

Substituting x = 4 and y = 9 into the equation, we get:

z = 15(9/4)

This equation now contains only one unknown, z, which we can easily solve for. The calculation involves multiplying 15 by the fraction 9/4. This step demonstrates the practical application of the joint variation equation, showcasing how it can be used to determine the value of one variable given the values of the others and the constant of variation.

Step-by-Step Solution for z

To find the value of z, we continue with the equation we established in the previous step:

z = 15(9/4)

First, we multiply 15 by 9:

z = 135/4

Now, we divide 135 by 4 to obtain the value of z:

z = 33.75

Therefore, when x = 4 and y = 9, the value of z is 33.75. This is the final answer to the problem, and it demonstrates the application of the principles of inverse and direct variation in a practical calculation. The value of z is directly influenced by the values of x and y, as dictated by the joint variation equation and the constant of variation. This solution highlights the power of mathematical relationships in modeling and predicting real-world scenarios.

Final Answer

The value of z when x = 4 and y = 9 is 33.75. This result is obtained by first establishing the joint variation equation, determining the constant of variation using the initial conditions, and then substituting the new values of x and y into the equation. This problem serves as a clear example of how mathematical concepts like direct and inverse variation can be applied to solve practical problems and understand relationships between variables.

Summary of the Solution Process

To recap, we solved this problem by following these key steps:

  1. Understanding the Concepts: We began by defining direct and inverse variation, which are crucial for setting up the problem correctly. Grasping these concepts allowed us to translate the problem statement into a mathematical equation.

  2. Establishing the Joint Variation Equation: We formulated the equation z = k(y/x) to represent the relationship between z, x, and y, where k is the constant of variation. This equation captures the essence of the joint variation, combining both direct and inverse relationships.

  3. Solving for the Constant of Variation (k): Using the given initial conditions (x = 6, y = 2, z = 5), we substituted these values into the equation and solved for k, finding that k = 15. The constant of variation is a critical element in defining the specific relationship between the variables.

  4. Substituting New Values and Calculating z: We then substituted the new values (x = 4, y = 9) and the calculated value of k into the equation to find z. This step demonstrates the practical application of the joint variation equation.

  5. Final Calculation: Through straightforward arithmetic, we determined that z = 33.75 when x = 4 and y = 9. This is the final solution to the problem, providing a concrete answer to the question posed.

This step-by-step approach not only leads to the correct answer but also provides a clear understanding of the underlying mathematical principles at play. By breaking down the problem into manageable steps, we can effectively apply these concepts to solve similar problems in the future. The combination of conceptual understanding and methodical calculation is key to success in mathematics.