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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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Most popular questions from this chapter

Fill in each blank with the correct response. (a) If the constant of variation is positive and \(y\) varies directly as \(x,\) then as \(x\) increases, \(y=\) ____. (increases/decreases) (b) If the constant of variation is positive and \(y\) varies inversely as \(x,\) then as \(x\) increases, \(V\) ____. (increases/decreases)

Solve each variation problem. For a constant area, the length of a rec tangle varies inversely as the width. The length of a rectangle is \(27 \mathrm{ft}\) when the width is \(10 \mathrm{ft}\). Find the width of a rectangle with the same area if the length is \(18 \mathrm{ft}\).

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{2}{3}+\frac{8}{27} $$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{m^{2}+3 m+2}{m^{2}+5 m+4} \cdot \frac{m^{2}+10 m+24}{m^{2}+5 m+6}$$

Solve each problem involving direct or inverse variation. If \(p\) varies inversely as \(q,\) and \(p=7\) when \(q=6,\) find \(p\) when \(q=2\)
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