Problem 61 - 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 \(y=3 .\) \(z=72\) when \(x=3\) and \(y=-6\)

Step-by-Step Solution

Answer

The function modelling the relationship is \(z = -4xy\), and the value of \(z\) when \(x = -2\) and \(y = 3\) is \(24\).

Step 1: Find k

Since z varies jointly with x and y, the formula can be expressed as \(z = kxy\), where k is the constant of variation. Given that \(z=72\), \(x=3\) and \(y=-6\), we can substitute these values into the equation to find \(k\), as such: \(72 = k * 3 * (-6)\). Solving for k gives \(k = -4\).

Step 2: Find z using the new k value

After establishing that \(k = -4\), we substitute this value and the values of x and y into our formula \(z = kxy\) to find z. Replacing \(x=-2\) and \(y=3\), the equation becomes \(z = -4*(-2)*3\). After calculating, the result for z is \(24\).

Key Concepts

Constant of Variation

In joint variation, when you see a phrase like "varies jointly," it often refers to a situation where a variable is influenced by more than one other variable at the same time. Here, the idea is similar to direct or inverse variation, but now, you have multiple influence factors. The constant of variation, symbolized as \( k \), is a crucial part of these equations. It allows you to turn a general relationship into a specific one that works with specific numbers.

To find \( k \), you start by having an equation based on the context of the problem—usually written as \( z = kxy \) when dealing with joint variation between \( z \), \( x \), and \( y \). Once you have enough specific values to substitute into this equation, such as \( z = 72 \), \( x = 3 \), and \( y = -6 \), you can solve for \( k \). In this exercise, the result was \( k = -4 \).

Understanding \( k \) gives you the unique key that models the relationship between your variables in that particular case. Once you have \( k \), you can predict or modify the equation depending on changes in \( x \) or \( y \). It's like finding the secret sauce to make the equation work in applicable scenarios.

Function Modeling

Function modeling is essentially about creating a mathematical formula that represents a particular relationship between variables. In context with this exercise, function modeling involves formulating an equation, \( z = kxy \), to describe the joint variation of \( z \) with \( x \) and \( y \). This equation encapsulates how these three values relate to each other under this specific type of variation.

You are turning real-world relationships—how things actually change together—into a mathematical format that can be better analyzed and used. The function \( z = kxy \) not only helps explain the initial scenario provided but also assists in making predictions about \( z \) when \( x \) and \( y \) change.

When using function modeling, always start by identifying what's varying and in relation to what. In this case, you have \( z \) as a function of two other variables \( x \) and \( y \) jointly. Understanding each component allows for a more straightforward application in calculation settings and provides insights into the quantitative dynamics at play.

Algebraic Equations

Algebraic equations are expressions set equal to one another, forming an essential tool in mathematical problem-solving. They allow relationships between variables to be precisely defined and explored. In our exercise related to joint variation, the algebraic equation \( z = kxy \) is key. It serves as the structure that holds together the variations of \( z \), \( x \), and \( y \).

When dealing with algebraic equations, especially in the context of variation, it's important to first isolate one of the variables to find its relationship with others—like finding \( k \) from \( z = kxy \). Once you have the parameters, you can substitute back into the equation to solve for unknowns under new conditions.

It's akin to working with a puzzle. Each piece (or variable) depends on its arrangement and relationship to others to form a coherent whole. By solving the equation given the constraints and known values, you can uncover other unknowns, just as we calculated \( z = 24 \) for \( x = -2 \) and \( y = 3 \). This powerful tool of algebra not only tackles theoretical math problems but also practical issues in daily life scenarios, providing clear pathways to solutions.