/*! 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 43 Simplify. (Assume all denominato... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 43

Simplify. (Assume all denominators are nonzero.) $$ 1 x+5+4 x-22 x-2-1 x+5 $$

Short Answer

Expert verified
The simplified expression is \(-18x + 8\).

Step by step solution

01

Identify Like Terms

To simplify the expression, look for terms that have the same variable. In our expression, these terms are the ones involving x: \(1x, 4x, -22x, -1x\). All other terms are constants without variables and can be grouped together.
02

Combine the 'x' Terms

Add or subtract the coefficients of the like terms involving 'x'. Start by listing them: \(1x, 4x, -22x, -1x\). Simplify: \(1 + 4 - 22 - 1\), which equals \(-18\). This gives us \(-18x\).
03

Simplify Constant Terms

Next, deal with the constant terms: \(5 - 2 + 5\). Simplify these: \(5 - 2 + 5 = 8\).
04

Form the Final Expression

Combine the result from Step 2 and Step 3 to form the final simplified expression. The simplified expression is \(-18x + 8\).

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.

Like Terms
In algebra, identifying like terms is crucial to simplifying expressions effectively. Like terms are those terms in an expression that have the same variables raised to the same powers. For example, in the expression
  • \(1x + 5 + 4x - 22x - 2 - 1x + 5\)
the terms involving 'x' such as \(1x, 4x, -22x, -1x\) are considered like terms.
Even though these terms have different coefficients, they can be combined because they share the common variable 'x'.
Constant terms like \(5, -2,\) and \(5\) are also considered like terms because they do not have any variables and can be grouped together.
Grouping like terms helps in organizing the expression for further simplification.
Combining Like Terms
Once like terms are identified, the next step in simplifying an algebraic expression is combining them. This involves adding or subtracting the coefficients of these terms.
When combining the terms in the expression
  • \(1x, 4x, -22x, -1x\)
we focus on their coefficients: \(1, 4, -22,\) and \(-1\).
  • Performing the operations: \(1 + 4 - 22 - 1\) yields \(-18\).
So, these terms can be combined to form \(-18x\).
Similarly, the constants \(5 - 2 + 5\) result in \(8\), when combined.
This results in a much simpler expression, \(-18x + 8\).
Combining like terms reduces the complexity of expressions, making them easier to work with.
Coefficient
The term coefficient is integral to understanding how to work with like terms. A coefficient is the numerical part of a term that is multiplied by the variable. In the terms
  • \(1x, 4x, -22x,\) and \(-1x\),
coefficients are \(1, 4, -22,\) and \(-1\) respectively.
To combine like terms, you add or subtract these coefficients while keeping the variable part unchanged.

It's important to note that if a term is just \(x\), it actually has a coefficient of \(1\), and if it's \(-x\) it has a coefficient of \(-1\).
Understanding coefficients is essential for any algebraic manipulation, including simplifying expressions, solving equations, and understanding polynomials.

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. $$3 \times 2=x+13-x$$

Solve for the indicated variable. Solve for \(w: P=2(l+w)\).

Solve for the indicated variable. Solve for \(\pi: r=C 2 \pi\).

A larger pipe fills a water tank twice as fast as a smaller pipe. When both pipes are used, they fill the tank in 5 hours. If the larger pipe is left off, then how long would it take the smaller pipe to fill the tank?

Solve. $$9 x 3 x-1-4 x=0$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.