Problem 26 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 7 a b+4 a b $$

Step-by-Step Solution

Answer

Answer: \(11ab\)

Step 1: Identify the like terms

We are given the expression: $$ 7ab + 4ab $$ Both terms are like terms since they have the same variables a and b.

Step 2: Combine the like terms

To combine the like terms, we add their coefficients (the numbers in front of each term): $$ (7+4)ab $$

Step 3: Simplify the expression

Now, we simplify the expression: $$ 11ab $$ The simplified algebraic expression is \(11ab\).

Key Concepts

Like Terms

Understanding like terms is crucial when working with algebraic expressions. Like terms are terms within an expression that have the same variable factors, meaning that they have the same variables raised to the same powers. For instance, in the expression

\(3x^2 + 5x^2 - 2x + 4\),

the terms \(3x^2\) and \(5x^2\) are like terms because they both contain the variable \(x\) raised to the second power. However, \(-2x\) and \(4\) are not like terms with \(3x^2\) or \(5x^2\) because they either have a different variable, a different power, or no variable at all.

In our original exercise, \(7ab + 4ab\) both terms are like terms because they contain the exact same variables, \(a\) and \(b\), each raised to the first power. There are no other variables or powers involved, making them compatible for the combining process.

Combining Like Terms

The process of combining like terms involves simplifying an algebraic expression by adding or subtracting the coefficients of like terms. The variables and their exponents do not change during this process. To combine like terms effectively, it’s important to:

• Identify the like terms
• Add or subtract their numerical coefficients
• Wrap up by rewriting the expression with the combined terms

In our example, to combine the like terms \(7ab\) and \(4ab\), we added their coefficients (7 and 4) to get:

\[(7 + 4)ab = 11ab\]

This step condensed the expression from two terms to one term without altering the variables or their exponents.

Algebraic Expression Simplification

Simplifying algebraic expressions is a basic yet essential skill in algebra. The goal is to make the expression as easy to understand as possible. To simplify an algebraic expression, you:

• Combine all like terms
• Use arithmetic to simplify coefficients and constants
• Keep the variables and their corresponding powers intact

Once all like terms are combined and the arithmetic is done, your expression should be in its simplest form. From the original exercise, after combining like terms, we got

\(11ab\),

which cannot be simplified further because it’s already at its most basic form with no additional like terms to combine. When algebraic expressions are fully simplified, it's easier to work with them in equations, inequalities, and various problem-solving scenarios.