/*! 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 85 Perform the indicated operations... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 85

Perform the indicated operations. \((3 x-1)-(10 x-6)\)

Short Answer

Expert verified
The result is \(-7x + 5\).

Step by step solution

01

Distribute the Negative Sign

First, distribute the negative sign across the terms in the second parenthesis. This changes the expression from \[(3x - 1) - (10x - 6)\] to \[(3x - 1) - 10x + 6\].
02

Combine Like Terms

Next, combine the like terms. The like terms are the terms with the variable \(x\) and the constant terms:- Combine \(3x\) and \(-10x\): \(3x - 10x = -7x\).- Combine \(-1\) and \(+6\): \(-1 + 6 = 5\).Now the expression simplifies to \[-7x + 5\].

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.

Distributing Negative Signs
When working with algebraic expressions, one crucial step is properly distributing negative signs. A negative sign in front of a parenthesis affects every term within those parentheses. To understand this better, let's break it down:
  • When you see a negative sign such as \((a - (b + c))\), imagine there is a "hidden" -1 being multiplied to everything inside the parenthesis.
  • Apply the distributive property, which means multiplying each term inside the bracket by -1. For example, \(-(b + c)\) becomes \(-b - c\).
So in the example \((3x - 1) - (10x - 6)\), distributing the negative sign turns it into \((3x - 1) - 10x + 6\).
This is because \(-(10x - 6)\) becomes \(-10x + 6\). Mastering this step is essential for simplifying algebraic expressions accurately.
Combining Like Terms
Once negative signs have been correctly distributed, the next step is to combine like terms. But what are like terms? Like terms are terms in an algebraic expression that have the same variables raised to the same power.
  • For example, \(3x\) and \(-10x\) are like terms because they both have the variable \(x\).
  • Similarly, constant terms like \(-1\) and \(+6\) are also considered like terms because neither of them have a variable attached.
To combine like terms, simply add or subtract their coefficients. In our expression \(3x - 1 - 10x + 6\), you combine:
  • \(3x\) and \(-10x\) to get \(-7x\).
  • \(-1\) and \(+6\) to get \(+5\).
This reduces the expression to \(-7x + 5\). Being able to identify and combine like terms is key to simplifying and solving algebraic expressions.
Simplifying Expressions
Simplifying expressions involves taking a potentially complex algebraic statement and rewriting it in the simplest form. This process includes distributing negative signs, combining like terms, and sometimes performing basic arithmetic.
Simplification is essential for understanding and solving mathematical problems, making them more manageable. Here's how it helps:
  • Reduces clutter: An expression like \(3x - 10x - 1 + 6\) becomes \(-7x + 5\) 鈥� much easier to interpret at a glance.
  • Aids in solving equations: Simplified equations are easier to manipulate when isolating variables or finding solutions.
In our example, by first addressing the parentheses and then combining like terms, we efficiently pared down the expression. The result \(-7x + 5\) represents the same relationship but is far more concise. Mastery of simplification strengthens overall problem-solving skills in algebra.

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

Although the actual amount varies by season and time of day, the average volume of water that flows over Niagara Falls (the American and Canadian falls combined) each second is \(7.5 \times 10^{5}\) gallons. How much water flows over Niagara Falls in an hour? Write the result in scientific notation. (Hint: 1 hour equals 3600 seconds.) (Source: niagarafallslive.com)

Add or subtract as indicated. $$ \left(2 x^{2}+7 x-9\right)+\left(x^{2}-x+10\right)-\left(3 x^{2}-30\right) $$

Subtract \(\left(4 y^{2}-6 y-3\right)\) from the sum of \(\left(8 y^{2}+7\right)\) and \((6 y+9)\)

Simplify each expression by performing the indicated operation. Explain how you arrived at each answer. a. \(z+3 z\) b. \(z \cdot 3 z\) c. \(-z-3 z\) d. \((-z)(-3 z)\)

Subtract using a vertical format. $$ \begin{array}{r} 7 a^{2}-9 a+6 \\ -\left(11 a^{2}-4 a+2\right) \\ \hline \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics 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.