/*! 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 75 Solve each equation. $$4-6(2 x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 75

Solve each equation. $$4-6(2 x-3)+1=3+2(5-x)$$

Short Answer

Expert verified
x = 1

Step by step solution

01

Expand and simplify both sides

First, expand the terms on both sides of the equation.The left side is:\[4 - 6(2x - 3) + 1 = 4 - 12x + 18 + 1 = 23 - 12x\]The right side is:\[3 + 2(5 - x) = 3 + 10 - 2x = 13 - 2x\]
02

Set the expanded sides equal and combine like terms

Now, set the simplified left and right sides equal to each other and get all the terms containing x on one side and constant terms on the other side of the equation:\[23 - 12x = 13 - 2x\]
03

Solve for x

First, add 12x to both sides to get:\[23 = 13 + 10x\]Then, subtract 13 from both sides to isolate the term with x:\[10 = 10x\]Finally, divide both sides by 10:\[x = 1\]

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.

Simplification of Equations
The first step in solving any linear equation is to simplify both sides of the equation. Simplification usually involves expanding and distributing any terms.

For instance, in our problem: $$4-6(2x-3)+1=3+2(5-x)$$, we need to distribute the -6 and the 2 into the parentheses.

This gives us:

- Left side: 4 - 6(2x - 3) + 1 becomes 4 - 12x + 18 + 1 = 23 - 12x

- Right side: 3 + 2(5 - x) becomes 3 + 10 - 2x = 13 - 2x

By doing this, we have simplified the original equation into easier terms.

Simplification helps to make the equation smaller and more manageable, minimizing errors and making it easier to see the next steps.
Combining Like Terms
After simplifying, the next step is combining like terms. Like terms are terms that contain the same variable raised to the same power.

In our equation, after simplifying, we have: $$23 - 12x = 13 - 2x$$

To combine like terms, we need to group the variable terms (terms containing 'x') and constant terms (just numbers) on opposite sides of the equation.

First, we add 12x to both sides:

- This moves the variable term $$-12x + 12x + 13 = -2x + 12x + 13$$Simplifying this gives,
$$23 = 13 + 10x$$

Combining like terms simplifies the equation to reduce the number of steps required. It also helps make subsequent steps, like isolating the variable, simpler to perform.
Variable Isolation
The final key step is isolating the variable. This process involves getting the variable you're solving for (in our case, x) alone on one side of the equation.

From our simplified equation:$$23 = 13 + 10x$$

Our next steps involve normal arithmetic operations:

    \t
  • First, subtract 13 from both sides:
    \t\t$$23 - 13 = 10x = 10x$$

    \t\tSolving that gives, \t
  • $$10 = 10x$$

    Next, divide both sides by 10:
    $$ \frac{10}{10} = \frac{10x}{10}$$
    This isolates *x*, leaving us
    $$ x = 1 $$


Each step brings the equation closer to the solution. Variable isolation makes the solving process clearer by breaking it down into simpler arithmetic steps.Understanding these key steps and concepts is crucial to mastering linear equations.
Each concept builds upon the previous points for an efficient and error-free solution.

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 each equation. Practice combining some steps. Look for more efficient ways to solve each equation. $$1-3(x-2)=4(x-1)-3$$

Solve each absolute value inequality and graph the solution set. See Examples 5鈥�7. $$-2|5-x| \geq-14$$

Show a complete solution to each problem. Missing integers. If the larger of two consccutive integers is subtracted from twice the smaller integer, then the result is \(21 .\) Find the integers.

Solve each problem. The area of a triangle is 40 square meters. If the height is 10 meters, then what is the length of the base?

Bachelor's degrees. The number of bachelor's degrees in thousands awarded in the United States can be approximated using the formula $$B=16.45 n+980.2$$ where \(n\) is the number of years since 1985 (National Center for Education Statistics, www.nces.ed.gov). What is the first year in which the number of bachelor's degrees will exceed 1.5 million?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics 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.