/*! 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 57 For Problems \(45-64\), perform ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 57

For Problems \(45-64\), perform the indicated operations. $$ 7 x+[3 x-(2 x-1)] $$

Short Answer

Expert verified
The expression simplifies to \(8x + 1\).

Step by step solution

01

Distribute Negative Sign

Start by addressing the expression inside the parentheses: \(3x - (2x - 1)\). Distribute the negative sign across \((2x - 1)\), turning it into \(-2x + 1\). Therefore, the expression simplifies to \(3x - 2x + 1\).
02

Simplify Inside Brackets

Simplify the expression \(3x - 2x + 1\). Combine like terms: \(3x - 2x = 1x\) or simply \(x\). Therefore, this reduces to \(x + 1\).
03

Replace and Simplify Entire Expression

Substitute \(3x - (2x - 1)\) with \(x + 1\) back in the main expression. The expression now becomes \(7x + (x + 1)\).
04

Combine Like Terms

Combine like terms in the expression \(7x + x + 1\). Add \(7x\) and \(x\) to get \(8x\). So, the expression becomes \(8x + 1\).

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.

Distributive Property
When working with expressions in elementary algebra, the distributive property becomes a critical tool. The distributive property involves distributing one expression over another inside a set of parentheses. It is expressed as:
  • \(a(b + c) = ab + ac\)
This means that if you have a negative sign or any term outside a parenthesis, it needs to be multiplied with every term within the parenthesis. For instance, in our exercise with the expression \(3x-(2x-1)\), the negative sign transforms into \(-1\), and we distribute it across both \(2x\) and \(-1\). This turns \(3x-(2x-1)\) into \(3x - 2x + 1\). It鈥檚 essential to correctly apply this rule, as it affects the integrity of the whole expression. Mistakes here can lead to incorrect results.
Combining Like Terms
Combining like terms is another key concept in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. Only the coefficients of these terms can differ. For example, if you have terms like \(3x\) and \(-2x\), they are like terms because both contain the variable \(x\).
  • The like terms must be combined by adding or subtracting their coefficients.
  • In the step \(3x - 2x\), you calculate the result as \(1x\) or simply \(x\).
Remember, combining like terms helps simplify expressions by reducing the number of separate terms. This skill is essential in algebra to ensure easier manipulation and understanding of algebraic expressions. Always scan for similar variables and powers.
Simplification Process
The simplification process in elementary algebra involves systematically reducing expressions to their simplest form. After distributing and combining terms, it's vital to continue this process to achieve the neatest expression possible.
  • Substitute simplified portions back into the main expression.
  • Further simplify by repeating the above steps as necessary.
In our example, after simplifying the part inside the parentheses to \(x + 1\), it was substituted back into the main expression, \(7x + (x + 1)\), which then gets combined to become \(8x + 1\). Each step should make the expression simpler and clearer, ready for any operations needed later. Simplification not only helps in reducing errors but also makes solving equations much more manageable.

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

For Problems \(82-112\), use one of the appropriate patterns \((a+b)^{2}=a^{2}+2 a b+b^{2},(a-b)^{2}=a^{2}-2 a b+b^{2}\), or \((a+b)(a-b)=a^{2}-b^{2}\) to find the indicated products. $$ (2-3 n)^{2} $$

For Problems \(11-36\), find the indicated products by applying the distributive property and combining similar terms. Use the following format to show your work: $$ \begin{aligned} (x+3)(x+8) &=x(x)+x(8)+3(x)+3(8) \\ &=x^{2}+8 x+3 x+24 \\ &=x^{2}+11 x+24 \end{aligned} $$ $$ (3 x-4)(7 x+1) $$

For Problems \(82-112\), use one of the appropriate patterns \((a+b)^{2}=a^{2}+2 a b+b^{2},(a-b)^{2}=a^{2}-2 a b+b^{2}\), or \((a+b)(a-b)=a^{2}-b^{2}\) to find the indicated products. $$ (x-4)^{2} $$

For Problems \(82-112\), use one of the appropriate patterns \((a+b)^{2}=a^{2}+2 a b+b^{2},(a-b)^{2}=a^{2}-2 a b+b^{2}\), or \((a+b)(a-b)=a^{2}-b^{2}\) to find the indicated products. $$ (x+9)^{2} $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (6 n-5)(2 n-3) $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.