/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 The relationship of \(x\) and \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 12

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=3, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=9\).

Short Answer

Expert verified
The constant of proportionality is 18. The equation is \( y = \frac{18}{x} \). When x=9, y=2.

Step by step solution

01

Determine the relationship

Since this is an inverse variation, the relationship between x and y can be defined as: \[ y = \frac{k}{x} \] where k is the constant of proportionality.
02

Find the constant of proportionality, k

Use the given values of x and y. When x = 3 and y = 6, substitute these values into the inverse variation equation to find k: \[ 6 = \frac{k}{3} \] Solve for k: \[ k = 6 \times 3 = 18 \]
03

Write the equation

Substitute the value of k back into the inverse variation equation: \[ y = \frac{18}{x} \]
04

Find y when x = 9

Substitute x = 9 into the equation to find y: \[ y = \frac{18}{9} \] Solve for y: \[ y = 2 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

constant of proportionality
Understanding the constant of proportionality is crucial to mastering inverse variation problems. This constant, often denoted as 饾憳, is a fixed number that links the two variables in an inverse variation relationship.
When two variables, say 饾懃 and 饾懄, vary inversely, their product is always the same. Mathematically, we express this as: \[ y = \frac{k}{x} \]
Here, 饾憳 is the constant of proportionality. To find 饾憳, you multiply the given values of 饾懃 and 饾懄. In our exercise, we are given that 饾懃=3 and 饾懄=6. Substituting these into the equation, we get: \[ 6 = \frac{k}{3} \]
Multiplying both sides of the equation by 3 to isolate 饾憳, we find: \[ k = 6 \times 3 = 18 \]
Thus, the constant of proportionality, 饾憳, is 18.
inverse variation equation
After determining the constant of proportionality, we can write the inverse variation equation that represents the relationship between 饾懃 and 饾懄. An inverse variation equation takes the form: \[ y = \frac{k}{x} \]
Given that 饾憳=18 from our calculation above, we substitute this value back into the equation, resulting in: \[ y = \frac{18}{x} \]
This equation tells us that for any value of 饾懃, we can find the corresponding value of 饾懄 by dividing 18 by 饾懃. This relationship is key in problems involving inverse variation.
solving for variables
Now that we have our inverse variation equation, solving for a variable is straightforward. Let's practice this with part (c) of our exercise: find 饾懄 when 饾懃=9. Using our inverse variation equation: \[ y = \frac{18}{9} \]
We simply substitute 9 for 饾懃 and solve the equation: \[ y = \frac{18}{9} = 2 \]
Therefore, when 饾懃=9, 饾懄=2.
To generalize, whenever you have an inverse variation equation and one of the variables, substitute the known value into the equation to find the unknown variable.
This method can be applied to any inverse variation problem, making it a versatile tool in algebra.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In radiography, a grid reduces the effect of X-ray scattering. The relationship of the interspace distance on the grid, \(x\), and the grid ratio, \(y\), is an inverse variation. When the interspace distance on a grid is 300 micrometers, the grid ratio is 8 . Write an equation that represents this variation. Include the units.

A student overdraws a bank account about five times each month. Predict the total overdraft fees the student will pay in 1 year. Chase's overdraft fees are \(\$ 25\) for the first fee each year, \(\$ 32\) for the next four and \(\$ 35\) after that. (Source: www.nytimes .com, Sept. 23, 2009)

For exercises \(67-82\), use the five steps and a proportion. Find the number of adults used to create the ratio "four out of five." Four out of five adults now use the Internet. 184 million adults are online from their homes, offices, schools or other locations. (Source: www.harrisinteractive.com, Nov. 17, 2008)

For a fixed length of household copper wire, the relationship of the cross- sectional area, \(x\), and the resistance, \(y\), is an inverse variation. When the cross-sectional area is \(3.14 \times 10^{-6} \mathrm{~m}^{2}\), the resistance is \(5.4 \times 10^{-3} \mathrm{ohm}\). a. Find the constant of proportionality, \(k\). Use scientific notation. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the resistance when the cross-sectional area is \(2.05 \times 10^{-6} \mathrm{~m}^{2}\). Round the mantissa to the nearest tenth.

For exercises 1-10, (a) solve. (b) check. $$ \frac{4}{15} k+\frac{3}{4}=-2 $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.