Problem 58 - Algebra 1

Question

The variables \(x\) and \(y\) vary directly. Use the given values of the variables to write an equation that relates \(x\) and \(y .\) $$x=9.8, y=3.6$$

Step-by-Step Solution

Answer

The equation that directly relates \(x\) and \(y\) is \(y=0.3673x\)

Step 1: Understanding Direct Proportionality

In a direct variation (or direct proportionality), the formula is usually of the form \(y=kx\), where \(k\) is the constant of proportionality.

Step 2: Calculate the Constant of Proportionality

Given \(x=9.8\) and \(y=3.6\), we can substitute these values into the equation \(y=kx\) to calculate the value of \(k\). Therefore, the formula becomes \(k=\frac{y}{x}\). Substituting, we get \(k=\frac{3.6}{9.8}\). Calculate the value of \(k\).

Step 3: Formulate the Direct Variation Equation

After calculating the constant \(k\), we can now substitute it into the \(y=kx\) equation to get the final result.

Key Concepts

Constant of Proportionality

In a direct variation, the relationship between two variables can be described with the equation \( y = kx \). Here, "\( k \)" is referred to as the **constant of proportionality**. This constant helps maintain the consistent ratio between the variables as they change. For example, if \( x = 9.8 \) and \( y = 3.6 \), you can find \( k \) by rearranging the formula: \( k = \frac{y}{x} \). Substituting the values gives \( k = \frac{3.6}{9.8} \).
This constant is crucial because it makes the relationship between the two variables easy to understand and predict. Any change in one variable can easily be translated into a change in the other using this same constant.

Equation Formulation

Once you have determined the constant of proportionality \( k \), you can easily formulate an equation that describes the relationship between \( x \) and \( y \). By substituting \( k \) back into the general form \( y = kx \), you can get the specific equation for the given values. If \( k \) calculated is a certain number, let's say \( 0.367 \), then the equation becomes \( y = 0.367x \).
This equation can now be used to understand, predict, and solve for either variable when one of the values is known. It is a precise tool for analysis and helps us describe how one variable directly affects another as they change, while keeping the ratio \( k \) consistent.

Proportionality Concept

The **proportionality concept** plays a pivotal role in understanding direct variation. Direct variation implies that two variables change in such a way that they maintain a constant ratio, as expressed through \( y = kx \). The central idea here is that as one variable increases, the other increases at a consistent rate, defined by the constant of proportionality.
Direct variation equations depict linear relationships, meaning if graphed, they will always yield a straight line passing through the origin. This straight line highlights the proportional nature because every point on the line maintains the same ratio \( k \). If you plot points using the derived equation \( y = 0.367x \), every point aligns in a consistent manner, proving the idea of direct proportionality.

• Proportionality ensures predictability and uniformity across the variables.
• It simplifies complex relationships by focusing on a constant ratio.

Understanding this concept is vital for recognizing patterns and making informed predictions based on the direct relationship of varying quantities.