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Chapter 6: Problem 19

Add or subtract the monomials. $$12 a+5 b-22 a$$

Short Answer

Expert verified
-10a + 5b

Step by step solution

01

Identify Like Terms

Look for terms that have the same variable raised to the same power. In this expression, the like terms are \(12a\) and \(-22a\).
02

Combine Like Terms

Add or subtract the coefficients of the like terms. For \(12a - 22a\), combine the coefficients to get \(-10a\).
03

Write the Final Expression

After combining the like terms, rewrite the expression. The final expression is \(-10a + 5b\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

like term
In algebra, a like term refers to terms that have the exact same variable part. This means the variables and their exponents must match. For example, in the given exercise, both terms having the variable 'a' (12a and -22a) are like terms because they share the same variable 'a.' Understanding like terms is critical because it enables you to simplify complex expressions by combining them.
Here are some more examples to illustrate:
  • 3x and 7x are like terms because they both contain the variable 'x.'
  • 4y^2 and -6y^2 are like terms because they both have 'y' squared.
  • However, 3x and 4y are not like terms because they contain different variables.
combine coefficients
After identifying like terms, the next step is to combine the coefficients of the like terms. The coefficient is the numerical part of the term, which means you will add or subtract these numbers.
In the exercise, for the like terms 12a and -22a, the coefficients are 12 and -22. You combine them by performing basic arithmetic operations. Note:
  • 12a - 22a = (12 - 22)a = -10a
Combining coefficients simplifies the expression and reduces the number of terms, making it easier to work with.
Remember always to keep track of positive and negative signs while performing these operations.
algebraic expressions
An algebraic expression is a combination of variables, numbers, and arithmetic operations (like addition and subtraction). In this exercise, we start with the expression 12a + 5b - 22a.
Here are key components:
  • Terms: Each part of the expression separated by + or - signs is a term (e.g., 12a, 5b, and -22a).
  • Variables: Symbols representing unknown values (e.g., a and b).
  • Coefficients: Numerical parts multiplying the variables (e.g., 12 in 12a).
By recognizing and combining like terms and coefficients, you simplify the algebraic expression. The final simplified expression in the exercise is: -10a + 5b. Understanding these components allows you to break down and solve more complex problems in algebra efficiently.

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Most popular questions from this chapter

Divide each polynomial by the monomial. $$\frac{63 a^{3}-108 a^{2}+99 a}{9 a^{2}}$$

Simplify. (a) \((-7)^{-2}\) (b) \(-7^{-2}\) (c) \(\left(-\frac{1}{7}\right)^{-2}\) (d) \(-\left(\frac{1}{7}\right)^{-2}\)

Divide each polynomial by the monomial. $$\frac{20 b^{2}-12 b}{-4}$$

Simplify. (a) \(m^{-4}\) (b) \(\left(x^{3}\right)^{-4}\)

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