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Chapter 7: Problem 15

Determine whether each equation represents direct or inverse variation. $$ y=\frac{12}{x^{2}} $$

Short Answer

Expert verified
The equation \( y = \frac{12}{x^2} \) represents inverse variation.

Step by step solution

01

Understand Direct and Inverse Variation

Direct variation occurs when one variable is a constant multiple of the other variable. Mathematically, it is represented as \( y = kx \) where \( k \) is a non-zero constant. Inverse variation occurs when one variable is a constant multiple of the inverse of the other variable. It is represented as \( y = \frac{k}{x} \) where \( k \) is a non-zero constant.
02

Compare Given Equation with Inverse Variation Formula

The given equation is \( y = \frac{12}{x^2} \). Notice that it has the form of \( y = \frac{k}{x^n} \) where \( k = 12 \) and \( n = 2 \). This resembles the form of an inverse variation equation.
03

Confirm Inverse Variation

Since the given equation fits the structure of \( y = \frac{k}{x^n} \), and because it represents an inverse relationship (as \( x \) increases, \( y \) decreases and vice versa), we can confirm that this is an equation of inverse variation.

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Key Concepts

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

Direct Variation
To understand direct variation, consider the mathematical formula: \( y = kx \). In this relationship, \( y \) changes directly with \( x \). If you increase \( x \), \( y \) will also increase in proportion, provided that \( k \) is a non-zero constant. For example, if you double \( x \), \( y \) will also double.

Here are some key characteristics of direct variation:
  • The graph of a direct variation equation is a straight line passing through the origin.

  • The ratio \( y/x \) remains constant (equal to \( k \)).
Understanding direct variation is crucial when comparing it to inverse variation. It helps to see where relationships between variables diverge.
Inverse Relationship
Inverse variation describes a situation where as one variable increases, the other decreases. The formula for inverse variation is \( y = \frac{k}{x} \), where \( k \) is a non-zero constant.

In this kind of relationship, when you multiply \( x \) by a factor, you must divide \( y \) by the same factor to keep \( k \) constant. For example, if \( x \) is doubled, \( y \) is halved.

Some essential properties of inverse variation include:
  • The product \( xy \) is always equal to the constant \( k \).
  • The graph of an inverse variation equation is a hyperbola, not passing through the origin.
The given equation \( y = \frac{12}{x^2} \) fits into the more general form \( y = \frac{k}{x^n} \), confirming the inverse relationship. This means as \( x \) increases, \( y \) decreases, demonstrating inverse variation.
Constant Multiple
The constant multiple \( k \) is a crucial component in understanding both direct and inverse variations. In any variation equation, \( k \) remains unchanged.

For direct variation \( y = kx \), \( k \) is the ratio of \( y/x \). For inverse variation \( y = \frac{k}{x^n} \), \( k \) is the product of \( yx^n \).

Let's highlight key points about constant multiples:
  • It provides the proportional relationship between variables.
  • In direct variation, \( k \) determines the slope of the line.
  • In inverse variation, \( k \) scales the hyperbola.
In our specific example \( y = \frac{12}{x^2} \), the constant multiple \( k \) is 12. This keeps the relationship between \( y \) and \( x^2 \) consistent. In inverse relationships, \( k \) helps guide how quickly \( y \) changes as \( x \) changes, providing vital insight into the relationship's nature.

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Most popular questions from this chapter

Solve each variation problem. The pressure exerted by water at a given point varies directly with the depth of the point beneath the surface of the water. Water exerts 4.34 lb per in. 2 for every 10 ft traveled below the water's surface. What is the pressure exerted on a scuba diver at \(20 \mathrm{ft} ?\)

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{3 t^{2}-4 t-4}{3 t^{2}+10 t+8} \div \frac{9 t^{2}+21 t+10}{3 t^{2}+14 t+15} $$

Rewrite each rational expression with the indicated denominator. $$ \frac{m-4}{6 m^{2}+7 m-3}=\frac{?}{12 m^{3}+14 m^{2}-6 m} $$

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{11}{6}-\frac{2}{5} $$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{2 m^{2}+7 m+3}{m^{2}-9} \cdot \frac{m^{2}-3 m}{2 m^{2}+11 m+5}$$
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