/*! 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 30 Solve each equation. See Example... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 30

Solve each equation. See Examples 6 and \(7 .\) $$ 14 x+7=7(2 x+1) $$

Short Answer

Expert verified
The equation is true for all real numbers; every \( x \) is a solution.

Step by step solution

01

Distribute Terms on the Right Side

The original equation is \( 14x + 7 = 7(2x + 1) \). Begin solving by distributing the 7 to both terms inside the parenthesis on the right side. This simplifies to: \( 14x + 7 = 14x + 7 \).
02

Simplify Both Sides

Compare both sides of the equation: \( 14x + 7 = 14x + 7 \). Since both sides are identical, no further simplification is needed.
03

Determine the Solution

Since the left side \( 14x + 7 \) is equal to the right side \( 14x + 7 \) for all values of \( x \), every real number is a solution to this 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
Equation solving is a fundamental skill in algebra and mathematics in general. It involves finding the value(s) of the variable(s) that make the equation true. In simpler terms, you are looking for what number or numbers you can substitute for the variable to balance the equation.
To solve equations efficiently, follow these steps:
  • Identify the type of equation you are dealing with. Linear equations, like in our example, have variables raised to the power of one.
  • Apply algebraic techniques such as addition, subtraction, multiplication, division, and distribution to both sides of the equation equally to maintain balance.
  • Simplify each side of the equation as much as possible.
  • Check your solution by substituting it back into the original equation to ensure it holds true.
In some cases, like the one we have here, you may find both sides are identical. This means the equation holds true for any value of the variable, concluding that the solution is every real number. Understanding these steps is crucial in mastering equation solving.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operators (like addition or subtraction) grouped together. They form the building blocks of equations and differ from equations as they do not have an equality sign.
Here's a quick breakdown of components of an algebraic expression:
  • Terms: Individual parts of the expression separated by plus or minus signs. In the expression \( 14x + 7 \), "14x" and "7" are terms.
  • Coefficients: The numerical part of a term that is multiplied by the variable. For "14x," the coefficient is 14.
  • Variables: Symbols used to represent unknowns, like \( x \).
Understanding algebraic expressions is essential to manipulate and solve equations. By analyzing each term, coefficient, and variable, you can apply the correct mathematical techniques to simplify expressions and solve equations.
Distributive Property
The distributive property is a key algebraic concept used to simplify expressions and solve equations. It states that multiplying a sum by a number is the same as multiplying each individual term by that number, then adding the results.
Take the expression \( 7(2x + 1) \) from our example. Using distributive property, this can be expanded to:
  • Multiply 7 by \( 2x \), giving \( 14x \).
  • Multiply 7 by 1, giving 7.
  • Combine these results to get \( 14x + 7 \).
This technique is especially useful when simplifying expressions with parentheses or when dealing with equations involving variables. By breaking down complex expressions into simpler terms, the distributive property helps you intuitively understand and solve equations.

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

The product of twice a number and three is the same as the difference of five times the number and \(\frac{3}{4}\). Find the number.

The measures of the angles of a particular triangle are such that the second and third angles are each four times the measure of the smallest angle. Find the measures of the angles of this triangle.

The flag of Brazil contains a parallelogram. One angle of the parallelogram is \(15^{\circ}\) less than twice the measure of the angle next to it. Find the measure of each angle of the parallelogram. (Hint: Recall that opposite angles of a parallelogram have the same measure and that the sum of the measures of the angles is \(360^{\circ} .\) )

Solve. For Exercises 1 through \(4,\) write each of the following as equations. The difference of three times a number and 1 is the same as twice the number. Find the number.

Solve each inequality. Write each answer using solution set notation. $$ 7(x-2)+x \leq-4(5-x)-12 $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.