Problem 4 - Algebra A Combined Function

Question

If y varies directly as \(x\), find the constant of variation and the direct variation equation for each situation. \(y=12\) when \(x=8\)

Step-by-Step Solution

Answer

The constant of variation is \( \frac{3}{2} \) and the direct variation equation is \( y = \frac{3}{2}x \).

Step 1: Understand Direct Variation

In a direct variation, two variables are related by the equation \( y = kx \), where \( k \) is the constant of variation. This means that as \( x \) increases or decreases, \( y \) does so proportionally. Our goal is to find the value of \( k \) based on the given values of \( y = 12 \) and \( x = 8 \).

Step 2: Find the Constant of Variation

We know that \( y = kx \). Substitute \( y = 12 \) and \( x = 8 \) into the equation to find \( k \). This gives us:\[12 = k \times 8\]To isolate \( k \), divide both sides by 8:\[k = \frac{12}{8}\]Simplify the fraction:\[k = \frac{3}{2}\]Thus, the constant of variation \( k \) is \( \frac{3}{2} \).

Step 3: Formulate the Direct Variation Equation

Now that we have \( k = \frac{3}{2} \), substitute \( k \) back into the direct variation equation to get:\[y = \frac{3}{2}x\]This is the equation that describes the direct variation relationship between \( y \) and \( x \).

Key Concepts

Constant of Variation

The constant of variation is a crucial part of understanding direct variation. When two variables, such as \( x \) and \( y \), vary directly, their relationship can be described by the equation \( y = kx \). Here, \( k \) is the constant of variation. This constant is the multiplier that relates the change in \( x \) to the change in \( y \). In simple terms, it tells us how much \( y \) changes when \( x \) changes by one unit. In our exercise, we were given that \( y = 12 \) when \( x = 8 \). By substituting these values into the direct variation equation, we found that \( k = \frac{3}{2} \).
This means that for every 1 unit increase in \( x \), \( y \) increases by \( \frac{3}{2} \) units. Knowing \( k \) helps us understand exactly how strongly \( x \) and \( y \) are linked in their variation.

• It determines the steepness of the relationship in a graph.
• A higher \( k \) value indicates a steeper change in \( y \) with respect to \( x \).

Direct Variation Equation

The direct variation equation provides a straightforward method to express the relationship between two directly varying variables. In mathematics, it is typically written as \( y = kx \) where \( k \) is the constant of variation. This equation captures the essence of a proportional relationship by describing how one variable changes with respect to another.
For example, when we discovered \( k = \frac{3}{2} \) in our scenario, substituting this back into the equation gave us \( y = \frac{3}{2}x \). This equation tells us that \( y \) is always \( \frac{3}{2} \) times the value of \( x \). In a graph, this relationship appears as a straight line that passes through the origin, showing that as \( x \) increases, \( y \) increases at a constant rate.

• The equation forms a linear relationship, characterized by a constant slope.
• Easy to predict \( y \) for any given \( x \) using this equation.

Proportional Relationship

A proportional relationship is one where two quantities change at a constant rate relative to each other. When one quantity increases, the other increases at the same rate, and vice versa. In the context of direct variation, this is precisely how the two variables \( y \) and \( x \) interact. Our direct variation equation, \( y = \frac{3}{2}x \), perfectly illustrates this proportional relationship.
Because the equation is linear and passes through the origin, every pair of \( x \) and \( y \) values will maintain the same ratio. This confirms that this relationship is proportional, meaning that the ratio \( \frac{y}{x} = k \) remains constant.

• Defines a linear relationship on a graph that consistently maintains proportion across values.
• A fundamental aspect of direct variation is its direct proportionality.

Understanding this helps not only in recognizing linear relationships but also in calculating how one variable affects another when they are proportional.