/*! 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 3 Simplify the algebraic expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

Simplify the algebraic expressions by combining similar terms. $$ 5 a^{2}-6 a^{2} $$

Short Answer

Expert verified
The simplified expression is \(-a^2\).

Step by step solution

01

Identify like terms

In the expression \(5a^2 - 6a^2\), identify the like terms. Both terms have the same variable part \(a^2\). So, they can be combined.
02

Subtract coefficients

Subtract the coefficients of the like terms. The coefficients are 5 and -6. Subtract these to combine the terms: \(5 - 6 = -1\).
03

Write the simplified expression

Now that we have the coefficient for the combined term, write the simplified expression as \(-1a^2\), or simply \(-a^2\).

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
When simplifying algebraic expressions, a critical first step is identifying **like terms**. Like terms are terms that have exactly the same variable component with the same exponents.
For instance, in the expression \(5a^2 - 6a^2\), both terms are "like" because they both contain \(a^2\).
Here are a few examples to help you better understand:
  • \(3x\) and \(5x\) are like terms (both have \(x\)).
  • \(7y^2\) and \(-2y^2\) are like terms (both have \(y^2\)).
  • \(4z\) and \(z\) are like terms (both have \(z\)); even if it鈥檚 written without a coefficient, like \(z\), it's understood to be \(1z\).
Recognizing like terms is essential as it allows you to simplify expressions by combining these terms effectively. You cannot combine terms with different variables or different powers, such as \(2x\) and \(2y\), since they are not like terms.
Coefficients
Understanding **coefficients** is another important part of simplifying algebraic expressions. A coefficient is simply the numerical factor in a term that includes a variable.
For example, in the term \(5a^2\), the coefficient is 5. Meanwhile, in the term \(-6a^2\), the coefficient is -6. Coefficients tell us how many units of a variable term are present.
Here are some examples to further illustrate:
  • In \(8y\), 8 is the coefficient of \(y\).
  • In \(-3x^2\), -3 is the coefficient of \(x^2\).
  • In \(z\), where no coefficient is explicitly written, the coefficient is 1 (since it鈥檚 really \(1z\)).
In algebra, coefficients are crucial for combining like terms, as it is the coefficients of these terms that you add or subtract, not the variable parts. This leads to a simplified expression.
Combining Terms
**Combining terms** is how we simplify algebraic expressions by merging like terms. When you combine like terms, you are essentially performing arithmetic on the coefficients while keeping the variable part the same.
Let's revisit our example: \(5a^2 - 6a^2\). Here, both terms have \(a^2\), making them like terms. To simplify, you subtract the coefficients: \(5 - 6 = -1\), leaving you with \(-1a^2\), or simply \(-a^2\).
Consider these additional examples:
  • \(3x + 7x = (3 + 7)x = 10x\)
  • \(2b^2 - 9b^2 = (2 - 9)b^2 = -7b^2\)
  • \(-4c + c = (-4 + 1)c = -3c\)
Combining terms is about focusing on the numerical coefficients. Maintain the integrity of the variables and powers. This method reduces expressions to their simplest form, making calculations easier and more streamlined.

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

Evaluate the algebraic expressions for the given values of the variables. $$ -4 x+9 y-3 x-y, \quad x=-4 \text { and } y=7 $$

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. Four less than one-half of a number

Evaluate the algebraic expressions for the given values of the variables. $$ \begin{aligned} &3\left(x^{2}-1\right)-4\left(x^{2}+1\right)-\left(2 x^{2}-1\right) \\ &x=\frac{2}{3} \end{aligned} $$

Answer the question with an algebraic expression. The product of two numbers is 98 , and one of the numbers is \(n\). What is the other number?

Answer the question with an algebraic expression. The perimeter of a rectangle is \(y\) yards and \(f\) feet. What is the perimeter expressed in feet?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.