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Chapter 1: Problem 1

Write an equation that expresses the statement. \(T\) varies directly as \(x\)

Short Answer

Expert verified
The equation is \( T = kx \), where \( k \) is a constant of proportionality.

Step by step solution

01

Identify the Direct Variation Statement

The statement says "\( T \) varies directly as \( x \)." In direct variation, we mean that when one quantity increases, the other one increases at a constant rate. Thus, \( T \) is directly proportional to \( x \).
02

Introduce the Constant of Proportionality

In direct variation, the relationship is expressed using a constant of proportionality, often denoted as \( k \). Thus, the relationship can be written as \( T = kx \).
03

Write the Final Equation

Using the information obtained, the final equation representing the direct variation of \( T \) with respect to \( x \) is \( T = kx \), where \( k \) is a constant.

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Key Concepts

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

Mathematical Equations
Mathematical equations are statements that assert the equality of two expressions. They are fundamental tools in understanding all sorts of patterns and relationships. In this case, we have
  • The equation \( T = kx \) which tells us how \( T \) changes in relation to \( x \).
  • In direct variation, equations help us understand how one variable moves in relation to another.
  • They are useful not just in mathematics, but also in real-world applications, such as physics and engineering.
Equations can be simple or complex, but they all aim to describe relationships or constraints in a precise mathematical notation.
For our direct variation example, the equation \( T = kx \) captures the linear growth pattern where the change in \( T \) is directly tied to the change in \( x \). By expressing this using an equation, we can predict how \( T \) behaves when \( x \) changes.
Proportionality Constant
The proportionality constant, often represented by \( k \), is a crucial part of direct variation equations. It tells us how strong the relationship between two variables is.
  • Think of \( k \) as the "rate" at which \( T \) changes per unit of \( x \).
  • When \( k \) is high, even a small change in \( x \) will cause a large change in \( T \).
  • If \( k \) is low, \( T \) will change more slowly as \( x \) increases.
Understanding this constant is key to grasping the essence of direct variation. In essence, the proportionality constant defines the slope of the line when we graph \( T \) against \( x \).
This slope tells us at a glance how quickly or slowly our dependent variable \( T \) is changing compared to its independent variable \( x \). So, when you solve problems involving direct variation, identifying the value of \( k \) will help you understand how the relationship works.
Algebra Concepts
In algebra, understanding direct variation is one of the foundational concepts that helps students grasp linear functions and relationships. These concepts are not only valuable in isolation but also create a basis for more advanced topics.
  • Direct variation is a type of linear relationship that can be expressed with the equation \( T = kx \).
  • This relationship is a perfect example of how variables are interconnected.
  • By mastering basic algebraic concepts like this, tackling more complicated relationships becomes significantly easier.
Getting comfortable with direct variation has several practical benefits. It allows students to
  • Predict outcomes of changes in variable values efficiently.
  • Graph and visualize data sets in meaningful ways.
  • Solve real-world problems where relationships can be captured simply.
In summary, algebraic concepts such as these form the backbone of mathematics, making it simple to analyze and leverage variable relationships in practical situations.

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Most popular questions from this chapter

Factor the expression completely. $$\left(a^{2}+1\right)^{2}-7\left(a^{2}+1\right)+10$$

Verify these formulas by expanding and simplifying the right-hand side. $$\begin{array}{l}A^{2}-1=(A-1)(A+1) \\\A^{3}-1=(A-1)\left(A^{2}+A+1\right) \\\A^{4}-1=(A-1)\left(A^{3}+A^{2}+A+1\right)\end{array}$$ Based on the pattern displayed in this list, how do you think \(A^{5}-1\) would factor? Verify your conjecture. Now generalize the pattern you have observed to obtain a factoring formula for \(A^{n}-1,\) where \(n\) is a positive integer.

Factor the expression completely. $$12 x^{3}+18 x$$

A square field in a certain state park is mowed around the edges every week. The rest of the field is kept unmowed to serve as a habitat for birds and small animals (see the figure). The field measures \(b\) feet by \(b\) feet, and the mowed strip is \(x\) feet wide. (a) Explain why the area of the mowed portion is \(b^{2}-(b-2 x)^{2}\) (b) Factor the expression in (a) to show that the area of the mowed portion is also \(4 x(b-x)\) CAN'T COPY THE GRAPH

Show that the equation represents a circle, and find the center and radius of the circle. $$x^{2}+y^{2}+6 y+2=0$$
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