Problem 12 - College Algebra

Question

Write an equation that expresses each relationship. Then solve the equation for y. \(x\) varies jointly as \(y\) and the square of \(z\)

Step-by-Step Solution

Answer

The equation expressing the given relationship is \(x = kyz^2\). When this is solved for \(y\), it gives \(y = \frac{x}{kz^2}\).

Step 1: Express the joint variation

First, express the given joint variation mathematically. When \(x\) varies directly as the product of \(y\) and the square of \(z\), we can write: \(x = kyz^2\). Here, \(k\) is the constant of variation.

Step 2: Solve the equation for y

Then, solve the equation for \(y\). To isolate \(y\), divide both sides of the equation by \(kz^2\). The result is: \(y = \frac{x}{kz^2}\). This equation expresses \(y\) in terms of \(x\), \(z\), and \(k\). Without specific values for \(x\), \(z\), and \(k\), we are limited to expressing \(y\) in terms of these variables.

Key Concepts

Direct Variation

In mathematics, direct variation describes a simple relationship between two variables in which one of the variables is a constant multiple of the other. This means that as one variable increases or decreases, the other does so as well, in direct proportion.

For example, if we say that x varies directly as y, this can be represented as \( x = ky \), where \( k \) is the constant of variation. This constant remains the same as x and y change. In the context of multiple variables, such as in the exercise involving variables x, y, and z, the situation is referred to as joint variation. Here, x varies directly as the product of y and the square of z (z^2), leading to the general formula \( x = kyz^2 \).

Understanding direct variation is crucial in many fields like physics, economics, and engineering, where this concept is commonly applied to describe relationships between different quantities.

Solving for y

The process of solving for y typically involves re-arranging an equation to get y on one side, making it the subject of the formula. This is often done to express y in terms of other variables and constants and is essential for many types of algebraic problems.

In the given exercise, to solve for y in the equation \( x = kyz^2 \), one must isolate y on one side. This is achieved by dividing both sides of the equation by the product of k and z^2, resulting in \( y = \frac{x}{kz^2} \). Now the equation explicitly defines y in relation to x, z, and the constant of variation k.

Being comfortable with solving for a variable is an important skill, not only in algebra but in solving equations across various scientific disciplines where isolating a variable is necessary to either find a specific solution or to reformulate equations for further analysis. It enables us to see the dependence of one variable on others within an equation.

Constant of Variation

The constant of variation, represented by the symbol k, is a fixed number that relates the variables in an equation of direct or joint variation. It essentially determines how quickly one variable will change with respect to another.

For example, in the joint variation equation from the exercise, \( x = kyz^2 \), the constant k affects how x varies with changes in y and z. If k is large, a small change in y or z could result in a large change in x. Conversely, if k is small, x will change less noticeably with changes in y or z.

This constant is central to the understanding of the variation and, without it, the nature of the relationship between the variables cannot be fully described. In practical situations, determining the constant of variation from empirical data allows for predictions and deeper insights into the behavior of related variables.