/*! 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 7 Write an equation that expresses... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 7

Write an equation that expresses the statement. 2 is proportional to the square root of \(y\).

Short Answer

Expert verified
The equation is \( 2 = k \sqrt{y} \).

Step by step solution

01

Understand Proportionality Statement

When we say that a quantity is proportional to another, it means that the two quantities change at a constant rate. Mathematically, this can be expressed using a constant of proportionality, denoted as \( k \).
02

Identify the Proportionality Relation

The statement given is "2 is proportional to the square root of \( y \)." In mathematical terms, this can be expressed as \( 2 = k \sqrt{y} \), where \( k \) is the constant of proportionality.
03

Rewrite the Equation

The goal is to write an equation that not only represents the described situation but also includes a proportionality constant. The equation from Step 2 is already correctly expressing this: \( 2 = k \sqrt{y} \). Nonetheless, you can express \( k \) explicitly by rearranging the equation.
04

Solving for the Constant (if needed)

In the exercise, it's not strictly necessary to solve for \( k \) since the equation just needs to express the statement. However, if given any additional information about \( y \), \( k \) could be solved. For instance, if \( y = 4 \), then \( k = 1 \).
05

State the Final Equation

The final equation that expresses the original statement is \( 2 = k \sqrt{y} \). This equation represents that 2 is directly proportional to the square root of \( y \) with \( k \) as the proportionality constant.

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.

Understanding the Square Root
The term 'square root' is a fundamental concept in mathematics. When you find the square root of a number, you are essentially looking for a value that, when multiplied by itself, gives you the original number. Simply put, if \(x = \sqrt{y}\), then \(x^2 = y\). This can be a crucial operation in understanding relationships and patterns in math problems.
  • The square root of a positive number always results in a positive number.
  • You will often see it written with the radical symbol \(\sqrt{}\).
  • For example, the square root of 9 is 3, because \(3 \times 3 = 9\).
In the context of our problem, we see that the square root of \(y\) is directly related to another number, which involves a specific mathematical relationship expressed through proportionality.
The Constant of Proportionality Explained
When dealing with proportional relationships, an important term you will encounter is the 'constant of proportionality.' It essentially translates a direct proportional relationship into a mathematical equation. The constant of proportionality, denoted by \(k\), shows how one quantity changes in relation to another.
  • If two quantities are proportional, their ratio remains constant.
  • For example, if \(2 = k \sqrt{y}\), \(k\) ensures the relationship between 2 and \(\sqrt{y}\) is maintained.
  • This means that for any change in \(\sqrt{y}\), the value of 2 changes proportional to it due to \(k\).
Understanding the constant is vital as it allows you to predict one quantity when the other changes, maintaining the proportional balance.
Formulating a Mathematical Expression
A mathematical expression is a combination of numbers, variables, and operators signifying a mathematical relationship. Expressions are the backbone of problem-solving as they allow us to translate real-world situations into readable and manipulatable forms. For instance, in our exercise, the problem is summarized as \(2 = k \sqrt{y}\).
  • This equation neatly encapsulates the relationship described in the problem.
  • It combines numbers, the concept of a square root, and the constant \(k\), to provide a complete algebraic summary.
  • Manipulating expressions helps in solving for unknowns, such as finding \(k\) if specific values of \(y\) are known.
Ultimately, equations form the language of mathematics, allowing us to succinctly communicate complex relationships.

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

Factor the expression completely. $$5 a b-8 a b c$$

Factor the expression by grouping terms. $$-9 x^{3}-3 x^{2}+3 x+1$$

Perform the indicated operations and simplify. $$\left(3 x^{2} y+7 x y^{2}\right)\left(x^{2} y^{3}-2 y^{2}\right)$$

Factor the expression completely. $$2 x^{3}+4 x^{2}+x+2$$

State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$\frac{x}{x+y}=\frac{1}{1+y}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and 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.