Problem 59 - Elementary Algebra

Question

Simplify. $$ -a b_{2}+a_{2} b-2 a b_{2}+5 a_{2} b $$

Step-by-Step Solution

Answer

The simplified expression is \(6 a_{2} b - 3 a b_{2}\).

Step 1: Analyze and Group Like Terms

First, identify the like terms in the expression. We have terms containing the factor \(a b_{2}\), which are \(-a b_{2}\) and \(-2 a b_{2}\). There are also terms containing the factor \(a^{2} b\), which are \(a_{2} b\) and \(5 a_{2} b\). Group these like terms together for simplification.

Step 2: Simplify Each Group of Like Terms

Simplify the groups obtained by combining their coefficients. For the terms \(-a b_{2}\) and \(-2 a b_{2}\), their sum is \((-1 - 2) a b_{2} = -3 a b_{2}\). For \(a_{2} b\) and \(5 a_{2} b\), their sum is \((1 + 5) a_{2} b = 6 a_{2} b\).

Step 3: Combine the Simplified Groups

After simplifying each group of like terms, combine them to get the simplified expression. Combine \(-3 a b_{2}\) and \(6 a_{2} b\) to result in the final simplified expression.

Key Concepts

like terms

In algebra, the concept of "like terms" is crucial for simplifying expressions. Like terms are those terms in an expression that have the same variables raised to the same powers. This means that both the type and number of variables must be identical, though the coefficients may differ.

In the exercise provided, identifying like terms was a necessary step. You had two sets of like terms:

• Terms with \( ab_2 \): a) \(-ab_2\) b) \(-2ab_2\)
• Terms with \( a^2b \): a) \(a^2b\)
• \(5a^2b\)

Grouping like terms together simplifies the process of combining coefficients later. Without recognizing like terms, the simplification could not proceed. Each group can then be worked on to consolidate the expression further.

combining coefficients

Combining coefficients is an essential skill in algebraic simplification once like terms have been identified. This process involves adding or subtracting the numerical coefficients of the same set of like terms. By doing this, we effectively condense those terms into a single expression.

In the exercise, after grouping the like terms, the next step was to combine their coefficients:

• For \(-ab_2\) and \(-2ab_2\), add the coefficients (\(-1\) and \(-2\) to get \(-3a b_2\).
• For \(a^2b\) and \(5a^2b\), add the coefficients (1 and 5) to get \(6a^2b\).

Notice how the variables and their exponents remain unchanged during the combination of coefficients. Only the numerical parts change, consolidating the terms for further simplification.

algebraic simplification

Algebraic simplification is the process of reducing expressions to their most concise form. This process makes expressions easier to read and understand, paving the way for solving equations or further algebraic manipulations. It involves

• Identifying and grouping like terms
• Combining their coefficients,
  as previously discussed
• Rewriting the expression

In this case, after you have combined the coefficients to simplify the groups:

• Began with \(-ab_2 + a^2b - 2ab_2 + 5a^2b\),
  reduced to \(-3ab_2 + 6a^2b\).

Through algebraic simplification, the expression was transformed from one with four separate terms into a streamlined version with just two. This reduced form often provides a clearer pathway for further analysis or solution finding.