Problem 62 - Algebra 2

Question

Suppose \(z\) varies jointly with \(x\) and \(y .\) Write a function that models each relationship. Find the value of \(z\) when \(x=-2\) and \(y=3 .\) \(z=32\) when \(x=0.1\) and \(y=8\)

Step-by-Step Solution

Answer

The value of \(z\) when \(x=-2\) and \(y=3\) is -240.

Step 1: Determine the constant of variation

Let's first find the value of constant \(k\) using the given values of \(x\), \(y\), and \(z\). By substituting \(z=32\), \(x=0.1\), and \(y=8\) into the formula \(z = kxy\), we have \(32 = k*(0.1*8)\). Solving for \(k\), we get \(k = 32 / 0.8 = 40\).

Step 2: Write the function

Having obtained the value of \(k\), we can write our function as \(z = 40xy\). This function describes the relationship between \(z\), \(x\), and \(y\).

Step 3: Calculate \(z\) for given \(x\) and \(y\)

Next, we use our function to find \(z\) when \(x=-2\) and \(y=3\), by substituting these values into the function. So, \(z = 40*(-2*3) = -240\).

Key Concepts

Constant of Variation

In the realm of joint variation, the constant of variation plays a crucial role. When a variable like \(z\) varies jointly with two other variables, \(x\) and \(y\), it means \(z\) is directly proportional to the product of \(x\) and \(y\). This relationship is described using a constant, often denoted as \(k\).

• What is \(k\)? It's a constant value that relates the three variables in a consistent manner.
• Finding \(k\): Given certain values of \(x\), \(y\), and \(z\), you can determine \(k\) by substituting these known quantities into the joint variation formula \(z = kxy\).

For example, substituting \(z = 32\), \(x = 0.1\), and \(y = 8\) into the formula, you solve for \(k\) and get \(k = 32 / 0.8 = 40\). Thus, knowing \(k\) allows you to predict the relationship between \(z\), \(x\), and \(y\) for any given values. This predictable pattern is fundamental to understanding joint variation equations in algebra.

Algebraic Functions

Algebraic functions like joint variation help model relationships between variables. A function such as \(z = kxy\) is a clear example of how variables can be connected through multiplication and constants.

• Understanding the Format: In our example, \(z\), \(x\), and \(y\) are variables, and \(k\) is the constant that holds the equation together.
• Relationships in Action: The joint variation equation shows that \(z\) changes as a direct result of multiplying \(x\) and \(y\) by the constant \(k\).

This type of function is powerful because it is used in various fields, such as physics and economics, to demonstrate how changes in one or more variables affect another. Further, writing the algebraic function \(z = 40xy\) provides a simple yet robust tool for problem-solving across different scenarios.

Problem-Solving

Problem-solving with joint variation involves using given data to calculate unknown variables. Once the function is established, solving problems becomes a systematic process.

• Step-by-Step Calculation: Use the function \(z = 40xy\) to determine \(z\) for any given \(x\) and \(y\).
• Application: For example, if you are asked to find \(z\) when \(x = -2\) and \(y = 3\), the substitution into the function looks like this: \(z = 40(-2 \times 3) = -240\).

Each step in problem-solving with joint variation reaffirms the relationship described by the function. With practice, these calculations become intuitive, allowing for efficient and accurate analysis of many practical situations.