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Chapter 7: Problem 116

In the following exercises, simplify using the distributive property. $$-9(3 a-7)$$

Short Answer

Expert verified
-27a + 63

Step by step solution

01

- Identify the Expression Inside the Parentheses

The expression inside the parentheses is \(3a - 7\)
02

- Apply the Distributive Property

To apply the distributive property, multiply each term inside the parentheses by -9. Distribute this multiplication across both terms inside the parentheses: \(-9 \times 3a\) and \(-9 \times -7\)
03

- Perform the Multiplication

Calculate each multiplication separately: \(-9 \times 3a = -27a\) and \(-9 \times -7 = 63\)
04

- Combine the Results

Combine the results of the distribution to form the simplified expression: \(-27a + 63\)

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Key Concepts

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

Simplifying Expressions
Simplifying expressions helps in achieving a cleaner, more compact form. This process often involves combining like terms鈥攖erms that have the same variables raised to the same power.

For example, in the expression \( -9(3a - 7) \), we use distributive property to make it simpler. You'll see that by distributing -9 to each term inside the parentheses, we get new, combined terms that are easier to handle.

Overall, simplifying is a vital skill in algebra. It can make complicated problems more understandable. Frequent practice will make it feel second nature.
Multiplication
Multiplication is one of the basic arithmetic operations and a key component of algebra. In the context of algebraic expressions, it often involves multiplying coefficients (numbers) and variables.

In our exercise, we needed to multiply \( -9 \) with each term inside the parentheses \( (3a - 7) \). Distributing -9 across the terms, we perform two multiplications:

\( -9 \times 3a = -27a \)
\( -9 \times -7 = 63 \)

When multiplying, be careful with negative signs. A negative times a positive is negative, and a negative times a negative is positive. This is crucial to get the correct result.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operators. They form the foundation of algebra. In our problem, \( 3a - 7 \) is a simple algebraic expression.

An important part of working with algebraic expressions is understanding how to manipulate them. This includes combining like terms, factoring, and applying properties like the distributive property.

In the given problem, we used the distributive property to simplify \( -9(3a - 7) \). By distributing \( -9 \) through the parentheses, we end up with a simple, combined expression: \( -27a + 63 \). The process of manipulating these expressions correctly is an essential skill in algebra.

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