Problem 11 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$ 14 a-3 b+5 b-6 a-b $$

Step-by-Step Solution

Answer

The simplified expression is \( 8a + b \).

Step 1: Identify Like Terms

Identify like terms in the expression. Like terms have the same variable raised to the same power. In the expression \( 14a - 3b + 5b - 6a - b \), the terms with \( a \) are \( 14a \) and \( -6a \), and the terms with \( b \) are \( -3b \), \( 5b \), and \( -b \).

Step 2: Combine Like Terms with \( a \)

Add or subtract the coefficients of like terms for \( a \). For \( 14a \) and \( -6a \), combine them: \[14a - 6a = (14 - 6)a = 8a\]

Step 3: Combine Like Terms with \( b \)

Add or subtract the coefficients of like terms for \( b \). For \( -3b \), \( 5b \), and \( -b \), combine them: \[-3b + 5b - b = (-3 + 5 - 1)b = 1b = b\]

Step 4: Write the Simplified Expression

Combine the simplified terms from Steps 2 and 3 to write the fully simplified expression:\[8a + b\]

Key Concepts

Simplifying Expressions

When dealing with algebraic expressions, simplifying them is a crucial skill that helps to reduce complexity. This process involves combining like terms, making the expression easier to understand and work with. To simplify an expression, look for terms that have the same variable parts. These are called "like terms". They can be combined by adding or subtracting their coefficients. By performing these operations, you consolidate the expression into a more manageable form. For example, in the expression \( 14a - 3b + 5b - 6a - b \), the like terms are those that contain the same variable: \( 14a \) can be combined with \( -6a \), and \( -3b \) can be combined with \( 5b \) and \( -b \). As you practice simplifying expressions, always remember:

• Identify like terms by looking for the same variable.
• Add or subtract the coefficients of like terms.
• Rewrite the expression with the simplified terms.

Simplifying expressions is a foundational skill in algebra and is essential for solving equations efficiently.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. They form the basis of algebra and allow us to represent real-world situations in mathematical terms. An expression does not have an equality sign like an equation does, so it cannot be "solved"—only simplified or evaluated.Consider the expression from our original exercise: \( 14a - 3b + 5b - 6a - b \). This expression contains variables \( a \) and \( b \), as well as numerical coefficients. Each term in the expression is a component that can be added or subtracted, providing us with a way to model different scenarios. The key components of an algebraic expression include:

• Variables: Letters that represent unknown values. In our example, 'a' and 'b' are the variables.
• Coefficients: Numbers that multiply the variables, like 14 and -6.
• Terms: Parts of the expression separated by plus or minus signs.

Understanding the structure of algebraic expressions is crucial for effectively manipulating them, especially when moving onto more advanced topics like equations and inequalities.

Polynomials

Polynomials are a specific type of algebraic expression that consist of terms in the form of \( ax^n \), where \( a \) is the coefficient and \( n \) is a non-negative integer exponent. They are used extensively in mathematics to describe a variety of relationships and are the building blocks for algebraic functions.In our exercise, the expression \( 14a - 3b + 5b - 6a - b \) can be viewed as a polynomial with terms linear in \( a \) and \( b \) (since both variables to the power of 1). When you combine like terms in a polynomial, you simplify it without changing its value. This is important because it often makes further algebraic operations, like addition or subtraction of polynomials, easier.Key characteristics of polynomials include:

• Degree: The largest sum of the exponents of variables in a single term. Linear polynomials have a degree of 1.
• Terms: Each polynomial is made up of multiple terms.
• Standard Form: Polynomials are often written with terms in decreasing order of their degrees.

Understanding polynomials and how to work with them is essential as it extends into various areas of mathematics, providing a solid foundation for calculus and beyond.