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When working with direct variation, the constant of variation, often denoted as \( k \), is a key component in understanding the relationship between two variables that change in proportion to each other. In simpler terms, it's the multiplier that relates the values of the variables.

To find the constant of variation, you need two corresponding values, one for each variable. In the given exercise, you have \( x = 54 \) and \( y = -9 \). By dividing \( y \) by \( x \), \( k = y/x \), you discover how much you need to multiply one variable to get the value of the other. In this case, \( k = -9/54 \) simplifies to \( k = -1/6 \). This means for every 6 units of increase in \( x \), \( y \) decreases by 1 unit, indicating an inverse proportionality between \( x \) and \( y \), which is unique to this direct variation relationship.

Direct variation encapsulates the idea of proportional relationships. When two variables have a constant ratio, that is, the ratio of one variable to another is consistent, they are said to be proportional to each other. This can be visualized as a straight line through the origin on a graph where one variable depends on the other.

From the example given, \( y = -1/6x \) represents a proportional relationship because for every increase or decrease in \( x \), \( y \) is affected in a predictably consistent way, based on the constant of variation. Understanding proportional relationships is crucial in subjects like physics and economics, where direct variation occurs frequently. For students, grasping this concept clarifies why equating ratios or setting cross-products equal to each other is a reliable method for solving problems involving proportions.

Once you have the constant of variation, the next step is writing the equation that will enable you to determine the value of one variable given the other. The formula for direct variation is simple: \( y = kx \), where \( k \) is the constant of variation. This equation confirms that \( y \) varies directly as \( x \).

Why is this simple equation important? With it, you can predict and calculate either variable knowing just one. In the exercise, once we know \( k = -1/6 \), the equation modelling the relationship is \( y = -1/6x \). This is an invaluable tool across many scientific and mathematical disciplines. It allows you to create models that can be analyzed and used to make predictions, which is the cornerstone of functions in algebra.