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In the given exercise, you are dealing with a direct variation between distance and the cost of gasoline. Direct variation means that as one value increases, the other value increases proportionally. This relationship can be expressed with the formula: \( y = kx \) where \( y \) is the dependent variable (cost of gasoline), \( x \) is the independent variable (distance driven), and \( k \) is the constant of proportionality. This structure shows that the cost of gasoline changes consistently with every mile driven. It's essential to note that in direct variation, the ratio between the two variables remains constant.

The constant of proportionality, denoted as \( k \), is a crucial element in the equation of direct variation. It helps us understand the fixed amount of change in one variable for every unit change in another variable. In this exercise, the constant of proportionality represents the cost per mile driven.To find \( k \), use the formula: \[ k = \frac{y}{x} \] Given the cost of gasoline is \(60 for 400 miles, we can calculate it as: \[ k = \frac{60}{400} = 0.15 \text{ dollars per mile} \] This means that for each mile driven, the cost of gasoline is \)0.15.

In problems involving distance and cost, it is common to find a direct variation relationship. The given exercise shows how the cost of gasoline is directly influenced by the distance driven. Understanding this relationship allows us to predict costs for various distances.In the example, we have: - Distance driven, \( x = 400 \text{ miles} \) - Cost of gasoline, \( y = 60 \text{ dollars} \) Using the known constant of proportionality (\( 0.15 \text{ dollars per mile} \)) in the equation \( y = 0.15x \), you can easily calculate the cost for any distance. For instance, to find the cost for driving 225 miles: \[ y = 0.15 \times 225 = 33 \text{ dollars} \] It is crucial to consider both distance and cost when planning trips and managing budgets effectively.

Linear equations form the foundation of direct variation problems. A linear equation is an equation between two variables that produces a straight line when plotted on a graph. In direct variation, the linear equation has the form: \( y = kx \) where \( k \) is the constant of proportionality.In the given exercise, the linear equation derived is: \[ y = 0.15x \] This equation helps predict the cost (\( y \)) for any given distance (\( x \)). Linear equations like these are crucial for understanding relationships in various fields, from economics to physics. They provide straightforward ways to model real-world scenarios, making them an essential tool in problem-solving.