Problem 113 - Intermediate Algebra

Question

a. \(6 a+6 a+6 a\) b. \(6 a+6 b+6 c\)

Step-by-Step Solution

Answer

(a) 18a (b) 6a + 6b + 6c

Step 1: Identify Like Terms (Part a)

The expression in part a is \(6a + 6a + 6a\). Identify any terms that are similar or like terms. In this case, all terms are alike because they all involve the variable \(a\).

Step 2: Add Like Terms (Part a)

Add the like terms together: \(6a + 6a + 6a = (6 + 6 + 6)a = 18a\). This simplifies the expression by combining the coefficients.

Step 3: Identify Like Terms (Part b)

The expression in part b is \(6a + 6b + 6c\). Identify any terms that are similar or like terms. In this case, none of the terms are alike because each involves a different variable: \(a\), \(b\), and \(c\).

Step 4: Expression Already Simplified (Part b)

Since none of the terms in \(6a + 6b + 6c\) are like terms, the expression is already simplified. It cannot be combined any further.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables are letters that represent unknown values and can vary, like \( a, b, \) or \( c \). An example of an algebraic expression is \( 3x + 5 \), which combines numbers and a variable.

These expressions do not have an equality or inequality sign, distinguishing them from equations or inequalities. Expressions, like the ones in the original exercise, can often be simplified by carrying out operations and combining like terms. The process of simplifying makes it easier to handle and use expressions in further calculations or equations. Understanding how to work with these expressions is a fundamental skill in algebra.

Like Terms

Like terms in algebra are terms that have identical variable parts, meaning their variables and their exponents are the same. For example, \( 5x \) and \( 3x \) are like terms because both have the variable \( x \), while \( 2x^2 \) and \( 4x^2 \) are also like terms due to matching variables and exponents.

Unlike terms, such as \( 6a \) and \( 6b \), cannot be combined through addition or subtraction because their variable parts differ. Recognizing like terms is essential for simplifying expressions effectively.

In the original exercise, part a with \( 6a + 6a + 6a \) could be combined because all terms were like (all containing \( a \)), whereas in part b \( 6a + 6b + 6c \) we couldn't combine because the variables differed.

Simplification of Expressions

Simplification of algebraic expressions involves combining like terms and performing basic arithmetic operations until no further simplification is possible. It helps to make expressions more manageable and clearer, often serving as a step towards solving equations.

For instance, in the expression \( 8y + 4 + 5y \), the like terms \( 8y \) and \( 5y \) are combined to \( 13y \), and the expression simplifies to \( 13y + 4 \).

In the original exercise, part a was simplified by combining three like terms \( 6a + 6a + 6a = 18a \). This made the expression cleaner and easier to interpret. However, part b \( 6a + 6b + 6c \) could not be simplified further, as the presence of different variables meant no terms shared the same variable and exponent combination.