/*! 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 the equation. $$12(2-x)=... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 30

Solve the equation. $$12(2-x)=6$$

Short Answer

Expert verified
The solution to this equation is \(x = 1.5\)

Step by step solution

01

Distribute Inside the Parentheses

First, distribute the 12 inside the parentheses to obtain a simplified equation: \(12 * 2 - 12 * x = 6\), which simplifies to \(24 - 12x = 6\)
02

Isolate the Term With the Variable

Next, isolate the term with the variable. To do this, subtract 24 from both sides of the equation to get: \(-12x = 6 - 24\) which simplifies to \(-12x = -18\)
03

Solve For the Variable

Finally, to solve for \(x\), divide both sides of the equation by -12. Thus, \(x = -18 / -12\), which simplifies to \(x = 1.5\)

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.

Distributive Property
The distributive property is a fundamental concept in algebra that allows you to simplify equations, making them easier to solve. It states that when you multiply a single term outside of parentheses by each term inside the parentheses, the equation remains equivalent. For example, in the equation \(12(2-x)=6\), applying the distributive property means you multiply 12 by both 2 and \(-x\).
  • Multiply 12 by 2 to get 24.
  • Multiply 12 by \(-x\) to get \(-12x\).
So, the equation \(12(2-x)=6\) turns into \(24 - 12x = 6\). By breaking down the equation in this manner, you eliminate parentheses and reduce complexity. This process helps set the stage for the next steps involved in solving equations.
Isolation of Variables
Isolation of variables is an important strategy used in solving equations, especially linear ones. The goal is to get the variable alone on one side of the equation. This makes finding its value straightforward. In our example, once the distributive property was applied, resulting in \(24 - 12x = 6\), the next step is to isolate \(-12x\).
  • Start by subtracting 24 from both sides to move constant terms to the opposite side of the equation.
  • This results in \(-12x = 6 - 24\).
  • The equation simplifies to \(-12x = -18\).
With \(-12x\) isolated on one side, you can clearly see what operations need to be performed in the next step to solve for \(x\). Remember, always perform the same operation on both sides to maintain the balance of the equation.
Solving Equations
Solving equations is the process of finding the value of the variable that makes the equation true. Once you've isolated the term containing the variable, the finishing touch is solving for \(x\). From the equation \(-12x = -18\), we need to isolate \(x\). This involves dividing both sides by \(-12\) to solve for \(x\).
  • Divide \(-18\) by \(-12\).
  • Performing the division gives \(x = 1.5\).
It鈥檚 crucial to note that the sign changes following the division, as dividing two negative numbers yields a positive result. Solving equations requires careful calculation to ensure each step logically follows the previous one, bringing you to the solution efficiently.

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 bill (parts and labor) for the repair of a car was S458. The cost of parts was \(\$ 339\). The cost of labor was \(\$ 34\) per hour. Write and solve an equation to find the number of hours of labor.

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$\frac{3}{4} \text { and }-\frac{5}{6}$$

\(2 \cdot 2 \cdot 2 \cdot 2\)

Apply the distributive property. $$2 y(8-7 y)$$

Simplify the expression. $$8\left(\frac{x}{8}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.