Problem 44 - Prealgebra

Question

Simplify expression. \(-3(a+2)-a\)

Step-by-Step Solution

Answer

The simplified expression is \(-4a - 6\).

Step 1: Distribute the Negative Sign

Begin simplifying the expression by distributing the negative sign in front of the parentheses. Multiply  through the terms inside the parentheses: \(-3(a+2) = -3 \cdot a + (-3 \cdot 2) = -3a - 6\). Thus, the expression becomes \(-3a - 6 - a\).

Step 2: Combine Like Terms

Next, identify and combine like terms in the expression \(-3a - 6 - a\). The like terms in this case are the terms containing 'a': \(-3a\) and \(-a\). Add these terms together to get \(-3a - a = -4a\). So, the new expression is \(-4a - 6\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental concept in prealgebra that involves reducing a mathematical expression to its simplest form. This process helps us understand and work with expressions more easily. In our example, we aim to simplify the expression \(-3(a+2)-a\).
To simplify, we need to apply various algebraic rules, including distributing and combining like terms. The goal is to rewrite the expression so that it contains as few terms as possible, making it more concise.

• The simpler an expression, the easier it is to solve equations or substitute values.
• Practicing simplification prepares you for more complex algebraic problems.

This skill is crucial as it makes further calculations easier and less error-prone.

Distributive Property

The distributive property is a key principle in algebra used to simplify expressions. It states that multiplying a number by a sum equals multiplying each addend separately and then adding those products together. In simple terms, you "distribute" the number outside the parentheses to each term inside the parentheses.
For the expression \(-3(a+2)\), we apply the distributive property:

• Multiply \(-3\) by \(a\) to get \(-3a\).
• Then, multiply \(-3\) by \(2\) to get \(-6\).

Thus, \(-3(a+2)\) simplifies to \(-3a - 6\).
Using the distributive property can simplify complex expressions and is especially useful in solving equations and real-world mathematical problems.

Combining Like Terms

Combining like terms is a method used to simplify expressions by merging terms that have the same variable part. This step is crucial to further simplify expressions after using the distributive property.
In the given example, the expression \(-3a - 6 - a\) contains like terms, \(-3a\) and \(-a\):

• Both terms have 'a' as their variable, making them like terms.
• We add them: \(-3a - a = -4a\).

By combining these terms, the expression simplifies to \(-4a - 6\).
This process reduces the number of terms, making arithmetic expressions simpler and easier to manage, especially when solving equations.