Problem 64 - Prealgebra

Question

Simplify each expression. $$7(a-2)$$

Step-by-Step Solution

Answer

The simplified expression is \(7a - 14\).

Step 1: Distribute the Factor

To simplify the expression \(7(a-2)\), we start by distributing the factor 7 to both terms inside the parentheses. This means we need to multiply 7 by \(a\) and 7 by \(-2\).

Step 2: Multiply and Simplify

First, multiply 7 by \(a\) to get \(7a\). Next, multiply 7 by \(-2\) to get \(-14\). Now, combine these results to form the simplified expression: \(7a - 14\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows us to multiply a single term across terms within a set of parentheses. Think of it as spreading the multiplication over all terms inside the brackets.
This property can be written as:

• For any numbers or variables, if you have a term like \(a(b + c)\), apply the distributive property by multiplying \(a\) with both \(b\) and \(c\):
• \(a(b + c) = ab + ac\)

To apply the distributive property correctly, ensure that each element inside the parentheses is multiplied by the external factor. This property helps break down expressions, making them easier to handle and solve.

Simplification

Simplification in algebra involves transforming an expression into a simpler, more efficient form, while maintaining its equivalence. The aim is to make the expression easier to work with or understand. Simplification follows a structured approach:

• First, use the distributive property to eliminate parentheses.
• Next, combine like terms, which are terms with the same variable component and exponent.
• Reevaluate the expression to ensure all possible simplifications have been completed.

This process can make solving equations or comparing expressions much clearer. Simplified expressions often provide intuitive insights into algebraic problems.

Multiplication of Terms

When multiplying terms in algebra, it's important to treat numbers and variables according to their properties. Multiplication follows key pathways:

• Multiply coefficients (the numerical parts) together first.
• Then, multiply like variables using exponent rules if necessary.

This is crucial when using the distributive property. For example, in the expression \(7(a - 2)\), multiply 7 with each term separately:

• Multiply 7 by \(a\) to yield \(7a\).
• Multiply 7 by \(-2\) to get \(-14\).

Keep each multiplication precise to ensure the final expression remains correct. Always combine your results to reflect the original intent of the expression.