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Chapter 0: Problem 44

Simplify each expression. $$ 6(a-5)-4(a+6) $$

Short Answer

Expert verified
The simplified expression is \( 2a - 54 \).

Step by step solution

01

Distribute the constants

Distribute the constants 6 and -4 to each term inside the parentheses: \[ 6(a-5) - 4(a+6) \rightarrow 6a - 30 - 4a - 24 \]
02

Combine like terms

Combine the like terms (terms with the variable 'a' and constant terms): \[ 6a - 4a - 30 - 24 \rightarrow 2a - 54 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a key concept in algebra that helps simplify expressions. It allows you to multiply a single term outside the parentheses by each term inside the parentheses. This process is essential when simplifying algebraic expressions.

In our exercise, we start with: \[ 6(a-5) - 4(a+6) \] \ Using the distributive property, we distribute the 6 to both 'a' and -5 and then distribute -4 to both 'a' and 6:
\[ 6(a-5) - 4(a+6) \rightarrow 6a - 30 - 4a - 24 \] \ This step breaks down complex expressions into simpler parts by eliminating the parentheses and making it easier to combine like terms later.
Combining Like Terms
Combining like terms is another fundamental step in algebraic simplification. It involves adding or subtracting terms that have the same variable and power. This makes the expression easier to manage.

In our exercise, after applying the distributive property, we have: \[ 6a - 30 - 4a - 24 \] \ Here, '6a' and '-4a' are like terms because they both contain the variable 'a'. Similarly, -30 and -24 are constants and can be combined:
\[ 6a - 4a - 30 - 24 \rightarrow 2a - 54 \] \ By combining like terms, we reduce the expression to '2a - 54', which is much simpler and more understandable.
Algebraic Simplification
Algebraic simplification involves using properties like the distributive property and combining like terms to transform complex expressions into simpler, more manageable forms. The goal is to make the expression as concise as possible.

Let's summarize our steps in the exercise:
  • Apply the distributive property: \[ 6(a-5) - 4(a+6) \rightarrow 6a - 30 - 4a - 24 \]
  • Combine like terms: \[ 6a - 4a - 30 - 24 \rightarrow 2a - 54 \]

By following these steps, we achieve the simplified form '2a - 54'.

Remember, breaking down expressions into smaller, manageable parts and then simplifying them step by step is a crucial skill in algebra. Practice makes perfect, so keep practicing!

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Most popular questions from this chapter

Write each percent as a fraction. Give answers in lowest terms. See Example 13. $$ 51 \% $$

Use an inequality symbol to write each statement. $$ 5+0 \text { is not equal to } 0 $$

Write each percent as a fraction. Give answers in lowest terms. See Example 13. $$ 7.5 \% $$

Simplify each expression. $$ 12+24 \div 3 \cdot 2 $$

When simplifying an expression, we usually do some steps mentally.Providing the property that justifies each statement in the given simplification. $$3 x+4+2 x+7$$ $$ =(3+2) x+(4+7) $$
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