Problem 8 - Algebra A Combined Function

Question

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=0.4\) when \(x=2.5\)

Step-by-Step Solution

Answer

The constant of variation is 0.16, and the equation is \( y = 0.16x \).

Step 1: Understand Direct Variation

Direct variation means that as one quantity increases, the other increases at a constant rate. This is mathematically expressed as \( y = kx \), where \( k \) is the constant of variation.

Step 2: Substitute Known Values

Given \( y = 0.4 \) when \( x = 2.5 \), substitute these values into the direct variation formula: \( 0.4 = k \times 2.5 \).

Step 3: Solve for Constant of Variation

To find \( k \), divide both sides of \( 0.4 = k \times 2.5 \) by 2.5: \( k = \frac{0.4}{2.5} \).

Step 4: Calculate the Constant

Perform the division: \( k = 0.16 \). This is the constant of variation.

Step 5: Write the Direct Variation Equation

Substitute the constant \( k = 0.16 \) back into the direct variation formula to get the equation: \( y = 0.16x \).

Key Concepts

Constant of Variation

In mathematics, the **constant of variation** is a crucial concept when discussing direct variation relationships. It serves as the coefficient of proportionality in the equation of direct variation, expressed as \( y = kx \). Here, \( k \) stands for the constant of variation.

The constant \( k \) indicates how much \( y \) changes for every unit change in \( x \). When given a direct variation problem, like \( y = 0.4 \) when \( x = 2.5 \), finding \( k \) involves substituting these values into the formula \( y = kx \) and solving for \( k \).

• If \( y = 0.4 \) and \( x = 2.5 \), the equation becomes \( 0.4 = k \times 2.5 \).
• To determine \( k \), divide both sides by 2.5, resulting in \( k = \frac{0.4}{2.5} = 0.16 \).

From this, we learn that \( k = 0.16 \) in this scenario. Once \( k \) is identified, it confirms how one variable scales with the other, setting the stage for constructing the full direct variation equation.

Proportional Relationship

A **proportional relationship** exists between two quantities when they increase or decrease at the same rate, maintaining a constant ratio. In mathematical terms, this relationship between \( y \) and \( x \) can be illustrated as \( y = kx \) where \( k \) is the constant of variation.

In our example, because \( y = 0.4 \) changes directly with \( x = 2.5 \), their relationship can be considered proportional. Each increase in \( x \) results in a predictable and fixed increase in \( y \) since the constant of variation \( k \) was calculated as 0.16.

This means that for every 1 unit increase in \( x \), \( y \) increases by 0.16:

• If you double \( x \), \( y \) doubles.
• If \( x \) is cut in half, \( y \) is halved too.

Understanding this proportionality is key to predicting and modeling real-world situations where one quantity depends consistently on another.

Mathematical Equation

In the realm of direct variation, a **mathematical equation** succinctly captures the relationship between two variables. This equation is formatted as \( y = kx \). Here \( y \) is the dependent variable, \( x \) is the independent variable, and \( k \) is the constant of variation.

Constructing the mathematical equation involves two primary steps:

• First, calculate the constant of variation \( k \) using given data points. For instance, \( y = 0.4 \) and \( x = 2.5 \) gives \( k = 0.16 \) through the calculation \( k = \frac{0.4}{2.5} \).
• Next, substitute this constant back into the formula \( y = kx \) to establish the specific equation connecting \( y \) and \( x \).

In our example, the final mathematical equation is \( y = 0.16x \). This allows us to express how \( y \) scales with \( x \) in a clear, concise, and predictable manner, making it a valuable tool for analyzing relationships in various contexts.