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Chapter 7: Problem 13

Determine whether each equation represents direct or inverse variation. $$ y=50 x $$

Short Answer

Expert verified
The equation represents direct variation.

Step by step solution

01

Understand Direct Variation

An equation represents direct variation if it can be written in the form \[ y = kx \] where \( k \) is a constant. This means as \( x \) increases, \( y \) increases proportionally.
02

Understand Inverse Variation

An equation represents inverse variation if it can be written in the form \[ y = \frac{k}{x} \] where \( k \) is a constant. This means as \( x \) increases, \( y \) decreases proportionally.
03

Compare the given equation with both forms

The given equation is \[ y = 50x \]. Compare this with the direct variation form \[ y = kx \]. In this case, \( k = 50 \). Thus, it matches the form of direct variation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Direct Variation
Direct variation describes a linear relationship between two variables. If you have an equation like \( y = kx \) where \( k \) is a constant, it鈥檚 a direct variation. This relationship means that as one variable increases, the other variable increases proportionally. For example, in the equation \( y = 50x \), when you increase \( x \), \( y \) also increases because \( k = 50 \). This means every time you multiply \( x \) by a certain number, \( y \) will increase by 50 times that number.

Some key points to remember about direct variation:
  • It always passes through the origin (0,0).
  • The ratio \(\frac{y}{x}\) is always constant and equal to \( k\).
  • You can easily identify it if the graph is a straight line through the origin.
Inverse Variation
Inverse variation describes a relationship where one variable increases as the other decreases. It can be written as \( y = \frac{k}{x} \) where \( k \) is a constant. This means the product of \( x \) and \( y \) is always constant.

For example, if you have \( k = 50 \) and you know \( x \) is 2, then \( y \) would be \( y = \frac{50}{2} = 25 \). If you increase \( x \) to 10, \( y = \frac{50}{10} = 5 \).

Some key things to note about inverse variation:
  • The graph of \( y = \frac{k}{x} \) is a hyperbola.
  • As \( x \) gets larger, \( y \) gets smaller, and vice versa.
  • The product \( xy \) is constant and equal to \( k \).
Constant of Variation
The constant of variation, \( k \), is a fixed number that relates the two variables in both direct and inverse variations.

In direct variation, the constant of variation is the multiplier \( k \) in the equation \( y = kx \). For example, in \( y = 50x \), \( k = 50 \). This tells you that for every 1-unit increase in \( x \), \( y \) increases by 50 units.

In inverse variation, \( k \) is the product of \( x \) and \( y \) in the equation \( y = \frac{k}{x} \). For instance, if \( y = \frac{50}{x} \), \( k = 50 \). This relationship means that as one variable increases, the other decreases, keeping the product constant.

Key points to remember about the constant of variation:
  • It remains the same regardless of the values of \( x \) or \( y \).
  • In direct variation, it is the ratio \( \frac{y}{x} \).
  • In inverse variation, it is the product \( xy \).

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