/*! 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 34 Solve each equation. See Example... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 34

Solve each equation. See Examples 6 and \(7 .\) $$ 2(x-5)=2 x+10 $$

Short Answer

Expert verified
No solution; the equation results in a contradiction.

Step by step solution

01

Distribute the Left Side

Distribute the 2 on the left side of the equation. Starting with: \[ 2(x-5) = 2x + 10 \] Distribute the 2:\[ 2 \times x - 2 \times 5 = 2x + 10 \]This simplifies to: \[ 2x - 10 = 2x + 10 \]
02

Eliminate Like Terms

Subtract \(2x\) from both sides to eliminate the \(2x\) terms:\[ 2x - 10 - 2x = 2x + 10 - 2x \]This simplifies to:\[ -10 = 10 \]
03

Analyze the Result

The equation \(-10 = 10\) is incorrect, indicating a contradiction. This means there is no solution, as this is not interpretable as a valid equation.

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
Solving equations is a core part of algebra. Equations often involve finding the value of a variable that makes the expression true. In algebra, we use a step-by-step process to simplify and solve these equations.
  • The goal is to isolate the variable on one side of the equation. This means you will perform operations like addition, subtraction, multiplication, or division on both sides to keep the equation balanced.
  • It's important to perform the same operation on both sides of the equation. This keeps the equation equal and valid throughout the solution process.
In our exercise, we are attempting to solve the equation \(2(x-5)=2x+10\). The process will reveal if the equation has a solution or not. Sometimes, the solution may result in a scenario where no values satisfy the equation, as you'll see later.
Distributive Property
The distributive property is a fundamental algebraic principle used to simplify equations. It allows you to multiply a single term by terms inside a parenthesis.
  • The property works by distributing the multiplication over addition or subtraction inside the parenthesis. Mathematically, it is expressed as \(a(b + c) = ab + ac\).
In the given exercise, the distributive property helps us break down \(2(x-5)\):
  • First, multiply the 2 by \(x\) and the 2 by \(-5\).
  • This results in \(2x - 10\), transforming the equation to \(2x - 10 = 2x + 10\).
This property is crucial for simplifying equations, making them easier to solve or analyze further.
No Solution Scenarios
Sometimes in algebra, you might encounter equations that, after simplification, don't make logical sense. These are known as 'no solution scenarios.'
  • A no solution scenario occurs when the two sides of an equation are always different numbers, meaning no possible value for the variable can make the equation true.
In our exercise, after distributing and simplifying, we ended up at an equation \(-10 = 10\). This equation is a contradiction; it cannot be true no matter what value is assigned to the variable. Hence, we conclude there is no solution.
  • Recognizing such scenarios is key because it tells us that either the initial conditions were flawed, or in real-world context, that perhaps the problem constraints are impossible to satisfy.

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

Twice the difference of a number and 8 is equal to three times the sum of the number and 3 . Find the number.

Evaluate each expression for the given values. See Section 1.8 \(r \cdot t ; \quad r=15\) and \(t=2\)

A 170 -pound person uses 5.3 calories per minute when bicycling at a speed of \(5.5 \mathrm{mph}\). How long must a person ride a bike at this speed in order to use at least 200 calories? Round up to the nearest minute. (Source: Same as Exercise 67)

Fill in the box with \(<,>, \leq,\) or \(\geq\). See the Concept Check in this section. If \(m \leq n,\) then \(-2 m \square-2 n\)

A 17 -foot piece of string is cut into two pieces so that the longer piece is 2 feet longer than twice the length of the shorter piece. Find the lengths of both pieces.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

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.