/*! 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 24 Solve the equation and simplify ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 24

Solve the equation and simplify your answer. $$-2 x+4=x-7$$

Short Answer

Expert verified
The solution is \( x = \frac{11}{3} \).

Step by step solution

01

Move 'x' terms to one side

We want to get all terms involving \(x\) on one side of the equation. Start by subtracting \(x\) from both sides:\[-2x + 4 - x = x - 7 - x\]This simplifies to:\[-3x + 4 = -7\]
02

Move constant terms to the other side

Now, we move the constant term 4 to the right side by subtracting 4 from both sides:\[-3x + 4 - 4 = -7 - 4\]Simplifying gives:\[-3x = -11\]
03

Solve for 'x'

To solve for \(x\), divide both sides by \(-3\):\[\frac{-3x}{-3} = \frac{-11}{-3}\]This simplifies to:\[x = \frac{11}{3}\]

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 Simplification
Equation simplification is like cleaning up a messy room. We take a complex expression and make it simpler or cleaner. In the context of linear equations, simplifying involves combining like terms and removing unnecessary terms. For example, if you have the equation \(-2x + 4 = x - 7\), simplifying it means rewriting and reorganizing the terms so they are easier to work with.

Here are some steps to simplify linear equations:
  • Combine like terms: These are terms with the same variable. For example, in our original equation, you find \(-2x\) and \(x\) in separate sides, which can be combined or moved together.
  • Isolate variables: This means getting all terms containing the unknown on one side of the equation. For our example, we bring \(x\) to one side by subtracting \(x\) from each side.
  • Reduce constants: Simplifying also involves combining constant numbers to make the equation as concise as possible. In our case, subtracting 4 from both sides.
Simplification sets the stage for efficiently solving the equation.
Solving Equations
Solving equations is like solving a puzzle. The goal is to find the value of the unknown variable that makes the equation true. For linear equations, this generally involves isolating the variable on one side of the equation.

With our example equation \(-2x + 4 = x - 7\), once simplified to \(-3x = -11\), solving involves a straightforward step:
  • Isolate \(x\): You need to make \(x\) stand alone. This is accomplished by dividing or multiplying to eliminate any coefficients or fractions.
  • Divide both sides: For \(-3x = -11\), divide each side by \(-3\) to obtain \(x\).
This yields the solution \(x = \frac{11}{3}\), the value of \(x\) that satisfies the equation.
Algebraic Manipulation
Algebraic manipulation refers to the rearrangement and balancing of terms in equations, using operations like addition, subtraction, multiplication, and division. This skill is crucial for solving and simplifying equations, helping to isolate the variable of interest.

In our example, we performed a series of manipulations:
  • Rearrange terms: We subtracted \(x\) from each side to consolidate the terms with \(x\) and the constants separately.
  • Balance equations: We subtracted the same value, 4, from both sides to maintain equality, a fundamental principle in algebra.
  • Constant coefficients: Finally, we divided by \(-3\) which was the coefficient of \(x\), to isolate the variable.
Mastering algebraic manipulation allows you to handle equations of increasing complexity with ease.

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

Solve the equation and simplify your answer. $$-6 x=-9 x+8$$

Solve the equation and simplify your answer. $$-\frac{1}{3} x+\frac{4}{5}=-\frac{9}{5} x-\frac{5}{6}$$

A trapezoid has bases measuring \(2 \frac{1}{4}\) and \(7 \frac{3}{8}\) feet, respectively. The height of the trapezoid is 7 feet. Find the area of the trapezoid.

Solve the equation and simplify your answer. $$\frac{5}{3} x+\frac{3}{2}=-\frac{1}{4}$$

Solve the equation and simplify your answer. $$-\frac{3}{4} x+\frac{2}{7}=\frac{8}{7} x-\frac{1}{3}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.