Problem 47 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ -6(4 a+3) $$

Step-by-Step Solution

Answer

The expression without parentheses is -24a - 18.

Step 1: Identify the structure of the given expression

The given expression is -6(4a+3), which consists of a number (-6) multiplied by a sum (4a + 3). This is a clear case for applying the distributive property.

Step 2: Apply the distributive property

To do this, -6 should be multiplied with each term inside the parentheses separately. That means -6 times 4a and -6 times 3.

Step 3: Simplify the expression

Multiply the terms identified in the previous step: -6 * 4a equals -24a and -6 * 3 equals -18.

Step 4: Write down the final expression

Combine the simplified terms back into one expression: -24a - 18.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are the backbone of algebra and are used to represent real-world problems in terms of numbers and variables. Think of them as a recipe that combines different ingredients (numbers, variables, and operations) to create something new.

An expression can include constants, like the number 3, and variables, like a, which stand in for unknown values. Operations such as addition, subtraction, multiplication, and division tie these elements together. For example, in the expression -6(4a + 3), -6 is the constant, a is the variable, and multiplication and addition are the operations at play.

Variables as Placeholders

Variables allow for generalizations in mathematics; they're like empty boxes waiting to be filled with a value. In the context of our expression, a can be any number, which means that the expression can represent many different numbers, depending on the value of a.

Combinations and Operations

Operations dictate how constants and variables interact. The parentheses in the expression indicate that the terms inside should be considered as one unit and that -6 multiplies the entire sum within them. By understanding these components and their interactions, you can begin to decipher and simplify complex algebraic expressions.

Simplifying Expressions Made Easy

Simplifying expressions is a way to make them easier to understand and work with by condensing them into a more manageable form without changing their value. Simplification is not just a neat trick; it is essential for solving equations, comparing expressions, and inevitably, succeeding in algebra.

The goal is to combine like terms and use operation properties to reduce the complexity of the expression. Consider the initial expression -6(4a + 3). Here, the simplification process involves distributing the multiplication over addition, eliminating the parentheses, and combining constants where possible.

Like Terms

When simplifying, look for like terms, which are terms that contain the same variables to the same power. In our example, there aren't any like terms to combine after using the distributive property. However, it's always a good step to check for, as combining like terms can further simplify your expression.

Remember that simplicity is key in algebra. Working with simpler expressions reduces the chance for errors and makes understanding the relationships between quantities clearer. The clearer the expression, the easier it will be to solve, interpret, and apply it to real-world scenarios.

Applying the Distributive Property

The distributive property is one of the most useful tools in algebra. It allows you to remove parentheses by distributing a factor across the terms inside those parentheses. It effectively 'shares out' the multiplication to each term.

In other words, if you have an expression like -6(4a + 3), the distributive property guides you to multiply -6 by 4a and -6 by 3 separately. This simplifies to -24a - 18, eliminating the need for parentheses altogether.

Steps to Distribute

The first step is to multiply the outside term by each term within the parentheses. It's crucial to maintain the signs of the numbers as you distribute. For instance, since we have a negative number outside the bracket, it's important to remember that a negative times a positive is a negative.

Clear and Error-Free

When done correctly, applying the distributive property will result in a clearer and often more concise expression. It's a surefire way to avoid errors down the line, especially when dealing with more complicated equations. Mastery of this property will give you a competitive edge in algebra, as it's frequently the first step in solving many algebraic problems.