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The constant of variation, often denoted as \( k \), is a crucial factor in direct variation equations. In direct variation, one variable changes consistently with another, maintaining a proportional relationship. Here, the equation \( W = kh \) describes how wages \( W \) vary in relation to hours worked \( h \). The constant \( k \) determines how the wage scales with each hour of work.

To find \( k \), you need a specific example. For instance, if someone earns \(1400 for a 40-hour week, you can insert these values into the equation to solve for \( k \):

• Equation: \( 1400 = k \cdot 40 \)
• Solve for \( k \): \( k = \frac{1400}{40} \), which simplifies to \( k = 35 \)

This means that for every hour worked, the wages increase by \)35. Thus, the constant of variation \( k \) is essentially the wage per hour.

Linear equations represent relationships with a constant rate of change and are foundational in understanding direct variation. In essence, a linear equation is any equation that graphs as a straight line. The direct variation equation \( W = kh \) is a linear equation. It reflects a straight-line relationship between wages and hours worked.

Linear equations can be expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. However, in the context of direct variation, the equation simplifies because the graph always passes through the origin, and the intercept \( b \) is zero:

• Direct variation form: \( y = mx \)
• Here, \( m \) (the slope) is the constant of variation \( k \)

This means the equation \( W = 35h \) describes Gloria's wages as a linear function of hours worked.

Understanding variable relationships is essential in direct variations as they highlight how one variable affects another. In the context of wages and hours, this relationship is straightforward: as the number of hours worked increases, wages increase proportionally by the constant of variation \( k \).

Such relationships are common in real-world scenarios where two variables maintain a consistent ratio. Since \( W \) directly depends on \( h \), any change in \( h \) will predictably change \( W \). For example:

• If Gloria works double the hours, her wages will also double, provided \( k \) remains constant.
• The formula \( W = 35h \) helps predict her income based solely on the hours reported.

This direct proportionality means that if she works 25 hours, calculate \( W = 35 \cdot 25 \) to find her earning, which is $875. This shows how important it is to understand the underlying linear relationship and the impact of the constant of variation.