Problem 6 - Prealgebra

Question

Use the distributive property to combine each of the following pairs of similar terms. $$9 a-a$$

Step-by-Step Solution

Answer

The combined expression is \(8a\).

Step 1: Identify Similar Terms

In the expression \(9a - a\), both terms involve the variable \(a\), making them similar terms. Here, \(9a\) means \(9 \times a\), and \(-a\) can be considered as \(-1 \times a\).

Step 2: Apply the Distributive Property

Factoring out the common variable \(a\), apply the distributive property: \(9a - a = a(9 - 1)\).

Step 3: Simplify Inside the Parentheses

Calculate the expression in the parentheses: \(9 - 1 = 8\). So, the expression simplifies to \(a \times 8\).

Step 4: Final Expression

Write the final simplified expression as \(8a\), which is the result of combining the like terms using the distributive property.

Key Concepts

Similar Terms

In algebra, a term is made up of numbers, letters, and sometimes a product of both, connected together to represent a mathematical value. When we talk about similar or like terms, we're referring to terms that have the same variable raised to the same power. For example, in the expression \(9a - a\), both terms include the variable \(a\) in the same degree, which makes them similar. This similarity is important because we can perform operations on these terms, like addition or subtraction. It's like combining apples with apples, not apples with oranges. Always look for terms with the exact same variables and exponents, so you know they can interact directly.

Combining Like Terms

Once you've identified similar terms, the next step is to combine them. Combining like terms involves adding or subtracting their coefficients. The coefficient is the number in front of the variable. For example, in \(9a - a\), the coefficients are 9 and -1, respectively. When combining these, you're essentially performing the operation \(9 + (-1) = 8\). Therefore, \(9a - a\) combines to become \(8a\). This simplification is key to making expressions more manageable and helps in solving equations more efficiently.

Prealgebra Concepts

Prealgebra lays the foundation for all future math courses by focusing on basic concepts like number operations, variables, and the order of operations. At this stage, understanding how to manipulate expressions using properties like the distributive property is crucial. The distributive property allows us to factor out common terms, simplifying expressions in the process. In the example \(9a - a\), recognizing the common factor \(a\) allows us to rewrite the expression using the distributive property as \(a(9-1)\), which then simplifies to \(8a\). This level of manipulation not only simplifies solving algebraic expressions but also develops logical thinking and problem-solving skills necessary for tackling more complex mathematical challenges in the future.