Problem 12 - Algebra 1: Concepts and Skills

Question

The variables \(x\) and \(y\) vary directly. Use the given values to write an equation that relates \(x\) and \(y .\) $$ x=3, y=9 $$

Step-by-Step Solution

Answer

The equation that relates \(x\) and \(y\) is \(y = 3x\).

Step 1: Understanding Direct Variation

In direct variation, the variables \(x\) and \(y\) are related by a constant of variation \(k\), such that \(y = kx\). Here, \(k\) is the constant of proportionality

Step 2: Find the Constant of Variation

Insert the given values into the equation. This is \(x = 3\) and \(y = 9\). So, \(9 = 3k\). By isolating the variable \(k\) on one side of the equation, divide both sides by 3 and then it gives \(k = 9/3 = 3\).

Step 3: Write the Equation of Variation

Substitute \(k\) into the equation \(y = kx\) to get the equation that relates \(x\) and \(y\). This leads to \(y = 3x\). This equation indicates that \(y\) is always three times as large as \(x\), which expresses the direct variation between the variables.

Key Concepts

Constant of Proportionality

The concept of the constant of proportionality is central to understanding direct variation. In a direct variation scenario, the constant of proportionality, often denoted by the letter \( k \), is the number that directly relates two variables. It essentially tells us how much one variable changes concerning the other. Think of it as a factor that keeps the variables in a consistent ratio. You can determine this constant by using the relationship equation \( y = kx \) and substituting the given values for the variables. In the exercise above, with \( x = 3 \) and \( y = 9 \), substituting these values into the equation yields \( 9 = 3k \). Solving this equation gives \( k = 3 \). This result indicates that for every unit increase in \( x \), \( y \) increases by three units, which is the essence of the proportionality constant in direct variation.

Equation of Variation

An equation of variation expresses the relationship between two variables that vary in relation to one another. In direct variation, this equation takes the form \( y = kx \), where \( k \) is, as mentioned earlier, the constant of proportionality. The main purpose of this equation is to allow prediction and calculation of one variable's value when the other variable changes, given the constant \( k \). For instance, from our exercise, once we know \( k = 3 \), the equation \( y = 3x \) can be used to find \( y \) for any given value of \( x \). This equation illustrates that \( y \) is three times the size of \( x \), establishing a clear and predictable pattern between the variables.

Variables Relationship

In the context of direct variation, the relationship between variables is straightforward and proportional. This means when one variable increases, the other does too, and they move together in a predictable manner. This predictable relationship is useful in many real-life scenarios, like calculating speeds, costs relative to quantities, and more. The derived equation \( y = 3x \) from our exercise tells us that \( y \) and \( x \) are directly linked through the proportionality constant 3. As a result, every incremental change in \( x \) will lead to a proportional, or threefold, change in \( y \). This direct relationship simplifies understanding how changes in one variable affect the other, making direct variation an essential concept in various fields of study and practical applications.