/*! 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 60 Solve. If no equation is given, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 60

Solve. If no equation is given, perform the indicated operation. $$ \frac{y}{3}=-4+y $$

Short Answer

Expert verified
The solution is \(y = 6\).

Step by step solution

01

Eliminate Fractions

Multiply both sides of the equation by 3 to eliminate the fraction: \(3 \times \frac{y}{3} = 3 \times (-4 + y).\) This simplifies to \(y = -12 + 3y.\)
02

Isolate Variable Terms on One Side

Subtract \(3y\) from both sides of the equation to get all the \(y\) terms on one side: \(y - 3y = -12.\) This simplifies to \(-2y = -12.\)
03

Solve for y

Divide both sides by \(-2\) to solve for \(y\): \(y = \frac{-12}{-2} = 6.\)

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.

Solving Linear Equations
Linear equations are fundamental in prealgebra and serve as building blocks for more complex mathematical concepts. The goal of solving a linear equation is to find the value of the variable that makes the equation true. In the original exercise, the equation given is \( \frac{y}{3} = -4 + y \). To solve this equation, we need to follow a systematic process.
  • Identify the variable: In this case, the variable is \( y \).
  • Perform algebraic operations to isolate the variable on one side of the equation.
  • Verify your solution by substituting the value back into the original equation.
In our provided step-by-step solution, we began by removing the fraction to simplify the process of isolating \( y \). By multiplying both sides by 3, we made it easier to deal with the equation. Once a solution is found, it is always wise to check if your answer satisfies the original equation to ensure accuracy.
Algebraic Manipulation
Algebraic manipulation is a key skill in solving linear equations. It involves using mathematical operations such as addition, subtraction, multiplication, and division to simplify and solve equations. In the exercise, after the fractions were eliminated, the equation was simplified to \( y = -12 + 3y \).Algebraic manipulation techniques include:
  • Like terms: Combine terms with the same variable parts.
  • Operations: Use arithmetic operations to simplify the equation.
  • Balancing: Always perform the same operation on both sides to maintain equality.
For our problem, after eliminating the fraction, we subtracted \( 3y \) from both sides. This helped move all terms involving \( y \) to one side, leading to the simplified equation \( -2y = -12 \). The consistent use of these operations ensures that the equation remains balanced, allowing us to isolate \( y \) and find its value.
Fractions in Equations
Handling fractions in equations often intimidates students, but with practice, these can be solved with ease. Fractions can best be managed by eliminating them early in the solving process. In the original equation, \( \frac{y}{3} = -4 + y \), the fraction was removed by multiplying both sides by 3.Key strategies for dealing with fractions:
  • Clear fractions early: Multiplying through by the denominator can simplify the equation.
  • Stay organized: Keep track of operations you perform, ensuring each step maintains the equation's integrity.
  • Equivalence: Ensure that operations are applied across the whole equation to maintain equality.
By clearing the fraction from the equation, you turn the problem into one with simpler arithmetic operations. This makes subsequent steps more familiar and straightforward, ultimately easing the path to the solution. Using these strategies consistently allows you to tackle equations involving fractions with confidence.

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

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{17}{5}$$

Answer true or false for each statement. It is possible for the difference of two fractions to be a whole number.

Perform the indicated operations. $$ -1 \frac{5}{7} \cdot\left(-2 \frac{1}{2}\right) $$

Perform the indicated operations. $$ -7 \frac{3}{10} \div(-100) $$

Solve. A student asked you to check his work below. Is it correct? If not, where is the error? \(3 \frac{2}{3} \cdot 1 \frac{1}{7} \stackrel{?}{=} 3 \frac{2}{21}\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.