Problem 22 - Intermediate Algebra

Question

Find the constant of variation for each of the stated conditions. \(y\) varies directly as \(x\) and inversely as \(z\), and \(y=24\) when \(x=36\) and \(z=18\).

Step-by-Step Solution

Answer

The constant of variation is 12.

Step 1: Understanding the Relationship

The problem states that \(y\) varies directly as \(x\) and inversely as \(z\). The mathematical representation of this relationship can be written as \(y = k \cdot \frac{x}{z}\), where \(k\) is the constant of variation.

Step 2: Substituting Known Values

We are given \(y = 24\), \(x = 36\), and \(z = 18\). Substitute these values into the equation \(y = k \cdot \frac{x}{z}\) to find \(k\). Thus, \(24 = k \cdot \frac{36}{18}\).

Step 3: Simplifying the Fraction

Simplify the fraction \(\frac{36}{18}\). Since \(36\) divided by \(18\) equals \(2\), the equation becomes \(24 = k \cdot 2\).

Step 4: Solving for the Constant of Variation

To find \(k\), divide both sides of the equation by \(2\). That gives \(k = \frac{24}{2}\). Calculating this gives \(k = 12\).

Key Concepts

Direct Variation

When something varies directly, it means an increase in one quantity results in a proportional increase in another quantity. This can be expressed simply as \( y = kx \), where \( y \) is the dependent variable, \( x \) the independent variable, and \( k \) the constant of proportionality. The constant \( k \) describes the rate at which \( y \) changes with respect to \( x \).
This relationship forms a straight line when graphed, passing through the origin, showcasing how direct variation essentially equates to a linear relationship.

• If \( k > 0 \), both \( y \) and \( x \) increase together.
• If \( k < 0 \), an increase in \( x \) leads to a decrease in \( y \).

In our exercise, this direct component comes with \( y = 24 \) when \( x = 36 \), helping set up the initial part of the equation.

Inverse Variation

Inverse variation implies a relationship where an increase in one quantity leads to a decrease in another. Mathematically, this is denoted as \( y = \frac{k}{z} \), where further increases in \( z \) result in decreases in \( y \), given a constant \( k \).
In graph terms, this produces a hyperbolic curve, illustrating how the variables move in opposition to each other.

• Similar to direct variation, the magnitude of change is defined by the constant \( k \).
• For inverse variation, \( y \) and \( z \) multiply to give a constant product \( k \).

In our example, as \( z = 18 \), it inversely affects the variation, demonstrating how changes in \( z \) adjust \( y \) based on that constant factor.

Algebraic Equations

Understanding algebraic equations is fundamental to solving variations. An algebraic equation is a mathematical statement of equality containing variables and constants. These equations serve as tools to express relationships between quantities.
In the solved problem, the given relationship \( y = k \cdot \frac{x}{z} \) includes both direct and inverse components, showing how equations can incorporate multiple variations.

• By substituting values into the equation, the task becomes to solve for the unknown, in this case, \( k \).
• This involves breaking down fractions and simplifying, ensuring clear use of known values to find constants.

Through step-by-step manipulation of the equation, one identifies how changes in \( x \) or \( z \) influence \( y \), reinforcing how algebraic equations reveal these nuanced mathematical relationships.