/*! 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 98 Determine whether each statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 98

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. Although I can solve \(3 x+\frac{1}{5}=\frac{1}{4}\) by first subtracting \(\frac{1}{5}\) from both sides, I find it easier to begin by multiplying both sides by \(20,\) the least common denominator.

Short Answer

Expert verified
The statement 'makes sense'. Multiplying each side of the equation by the least common denominator to begin with can simplify the process, especially for those who prefer not to work with fractions. It is alternative and acceptable way to solve the equation, however subtracting \(\frac{1}{5}\) from both sides first is equally valid and can be easier for some.

Step by step solution

01

Evaluate the original equation

Consider the original equation \(3x + \frac{1}{5} = \frac{1}{4}\). The aim is to isolate x to solve the equation, which often involves simplifying the equation.
02

Assess the proposed operation

The proposed operation is multiplying both sides by 20, the least common denominator of 5 and 4 (the denominators in the equation). If we perform this operation, we get \(20*(3x + \frac{1}{5}) = 20*\frac{1}{4}\), which simplifies to \(60x + 4 = 5\). Note that this operation eliminates the fractions, which can make the equation easier to solve.
03

Compare the proposed method with the alternative

An alternative operation is subtracting \(\frac{1}{5}\) from both sides first as hinted at in the statement. This gives us \(3x = \frac{1}{4} - \frac{1}{5}\), which simplifies to \(3x = \frac{1}{20}\). While it's not incorrect, this method requires dealing with fractions, which some people might find more challenging.

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.

Equation Solving
When it comes to solving equations, the main goal is to find the value of the unknown variable, often denoted as \(x\) or \(y\). This involves performing a series of operations that maintain the equality of both sides of the equation.
  • Start by simplifying both sides of the equation if necessary. This might involve distributing or combining like terms.
  • Next, use operations such as adding, subtracting, multiplying, or dividing to isolate the variable on one side of the equation.
    • In our case, we started with the equation \(3x + \frac{1}{5} = \frac{1}{4}\). The ultimate goal was to isolate \(x\), leading us to use operations like multiplication to simplify the process.
    Choosing the right operation depends on the equation's form and the involved numbers. As you'll see, understanding fractions and how to manipulate them can be crucial in equation solving.
    Fractions
    Fractions represent a part of a whole. In algebra, they often appear in equations and can pose a challenge if one is not familiar with handling them. Let's break it down simply.
  • A fraction consists of a numerator (top number) and a denominator (bottom number).
  • The key operations with fractions include addition, subtraction, multiplication, and division.
    • In our exercise, we're dealing with fractions: \(\frac{1}{5}\) and \(\frac{1}{4}\).
    To solve the equation efficiently, it helps to be comfortable with fractions so that operations like finding a common denominator or performing arithmetic with fractions become manageable.
    Least Common Denominator
    Finding the least common denominator (LCD) is a critical skill when dealing with fractions in equations. The LCD is the smallest multiple that two or more denominators share.
  • It simplifies the task of adding, subtracting, or comparing fractions.
    • In our equation, the denominators were 5 and 4. The LCD of these numbers is 20 because it's the smallest number that both 5 and 4 can divide without leaving a remainder.
  • Multiplying each side of the equation by the LCD helps to eliminate fractions altogether, which simplifies the solving process.
    • Applying this concept to our equation made it straightforward by transforming the fraction-laden equation to \(60x + 4 = 5\), a much easier form to handle.
    Isolating Variable
    The concept of isolating the variable is a central aspect of solving equations. Once the variable is isolated, the equation effectively tells us what number it equals.
  • The process involves rearranging the equation so that the variable stands alone on one side of the equality.
    • In the exercise, post multiplication by 20, our equation was \(60x + 4 = 5\). To isolate \(x\), we followed these steps:
    1. Subtract 4 from both sides to get \(60x = 1\). 2. Divide by 60 to solve for \(x\), yielding \(x = \frac{1}{60}\).
    This step-by-step simplification is typical in algebraic problems and helps to ultimately uncover the value of the variable sought.

    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

    Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The rate for a particular international telephone call is \(\$ 0.55\) for the first minute and \(\$ 0.40\) for each additional minute. Determine the length of a call that costs \(\$ 6.95\)

    On two examinations, you have grades of 86 and 88 . There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of \(\mathrm{A}\), meaning a final average of at least 90 . a. What must you get on the final to earn an A in the course? b. By taking the final, if you do poorly, you might risk the \(B\) that you have in the course based on the first two exam grades. If your final average is less than \(80,\) you will lose your \(B\) in the course. Describe the grades on the final that will cause this to happen.

    Evaluate: \(-10^{2}\)

    Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. $$4(2 y-1)>12$$

    An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?
    See all solutions

    Recommended explanations on Math Textbooks

    Decision Maths

    Read Explanation

    Theoretical and Mathematical Physics

    Read Explanation

    Calculus

    Read Explanation

    Pure Maths

    Read Explanation

    Mechanics Maths

    Read Explanation

    Geometry

    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.