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Chapter 7: Problem 13

Determine whether each equation represents direct, inverse, joint, or combined variation. $$ y=3 x z^{4} $$

Short Answer

Expert verified
Joint variation

Step by step solution

01

Identify the Variables

Identify the variables involved in the equation. Here, the variables are: \(y\), \(x\), and \(z\).
02

Analyze the Equation

Write the given equation. The equation is \(y = 3xz^4\).
03

Identify the Variation Type

Examine how the variables are related to one another. In the equation \(y = 3xz^4\), \(y\) is directly proportional to \(x\) and \(z^4\). There is no division of variables involved here.
04

Determine the Type of Variation

Since \(y\) changes based on the product of \(x\) and \(z^4\), it indicates a joint variation. Joint variation occurs when a variable depends on the product of two or more other variables.

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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 the relationship between two variables where one variable is a constant multiple of the other. Imagine you have a variable y and another variable x. In direct variation, as x increases, y increases proportionally, and as x decreases, y decreases proportionally. The formula for direct variation is given by the equation:

\( y = kx \),
where \( k \) is the constant of proportionality. Here are some key points about direct variation:
  • The graph is a straight line passing through the origin.
  • The ratio \( \frac{y}{x} \) is always constant.
  • If you double x, y will also double if they vary directly.
To illustrate, if you have a situation where \( y = 5x \), then you know every time x increases by 1, y increases by 5.
inverse variation
Inverse variation occurs when one variable increases while the other decreases such that their product is constant. In mathematical terms, if y varies inversely as x, the relationship is defined by the equation:

\( y = \frac{k}{x} \),
where \( k \) is the constant of inverse variation. Some important features of inverse variation include:
  • The product \( xy \) is always constant.
  • The graph of inverse variation is a hyperbola.
An example of inverse variation is the relationship between speed and travel time for a fixed distance. If you drive faster (increase speed), the time it takes to travel the same distance decreases, and this rate forms a hyperbola.
proportionality
Proportionality covers both direct and inverse variation. It is the concept that two quantities change in a dependent manner. When dealing with direct proportionality, as one quantity increases, the other also increases at a constant rate.

In the case of inverse proportionality, as one quantity increases, the other decreases, and their product remains constant. The equations for proportionality are:
  • For direct proportionality: \( y = kx \)
  • For inverse proportionality: \( y = \frac{k}{x} \)
Understanding proportionality helps you predict how changes in one variable affect another, making complex problems simpler to solve.
product of variables
The product of variables plays a crucial role in understanding different types of variation, including joint variation. Joint variation is when a variable depends on the product of two or more variables. For example, in the equation \( y = 3xz^4 \), you can see that y changes based on the product of x and \(z^4\). The important features to note here are:
  • Several variables multiply together to affect the outcome.
  • There is no division involved; it only involves multiplication.
  • The constant 3 in \( y = 3xz^4 \) indicates how much y changes for a unit change in the product of x and \( z^4 \).
Joint variation can seem complex, but breaking it down into how each variable contributes to the product can simplify understanding.

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Most popular questions from this chapter

Solve each problem. See Examples \(1-7\) The amount of light (measured in foot-candles) produce. by a light source varies inversely as the square of the distance from the source. If the illumination produced \(1 \mathrm{m}\) from a light source is 768 foot-candles, find the illumination produced \(6 \mathrm{m}\) from the same source.

Multiply or divide as indicated. $$ \frac{12 x-20}{5 x} \cdot \frac{6}{9 x-15} $$

Solve each problem mentally. Use proportions in Exercises 23 and 24. A van traveling from Atlanta to Detroit averages \(50 \mathrm{mph}\) and takes 14 hr to make the trip. How far is it from Atlanta to Detroit?

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{11 x-13}{2 x-3}-\frac{3 x-1}{2 x-3} $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{5}{t^{4} u^{7}}-\frac{3}{t^{5} u^{9}}+\frac{6}{t^{10} u} $$
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