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Chapter 1: Problem 147

Write an equivalent expression using the distributive law. $$ -5(2 a+3 b) $$

Short Answer

Expert verified
\(-10a - 15b\).

Step by step solution

01

Understand the Distributive Law

The distributive law states that for any numbers a, b, and c, the expression \(a(b + c) = ab + ac\), can be expanded by multiplying a with each term inside the parentheses individually.
02

Apply the Distributive Law

In the given expression \(-5(2a + 3b)\), distribute \(-5\) to both \(2a\) and \(3b\). So, \(-5 \cdot 2a\) and \(-5 \cdot 3b\).
03

Perform the Multiplications

Multiply \(-5\) by \(2a\) to get \(-10a\). Then, multiply \(-5\) by \(3b\) to get \(-15b\).
04

Combine the Results

Combine the results from the previous step to write the final expression: \(-10a - 15b\).

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. For example, in the expression \(-5(2a + 3b)\), \(-5\) is a constant, \(a\) and \(b\) are variables, and the entire term inside the parentheses is being multiplied by the constant \(-5\). Algebraic expressions are fundamental in algebra, allowing us to represent real-world scenarios where quantities vary. To work with these expressions, we often need to understand how to simplify or expand them, as seen in our problem exercise.
Distributive Law
The distributive law is a key property in algebra and is used to expand expressions. This law states that \(a(b + c) = ab + ac\). Essentially, you must multiply the term outside the parentheses by each term inside the parentheses individually. Consider the given expression \(-5(2a + 3b)\). According to the distributive law, \(-5\) should be multiplied by \(2a\) and \(3b\) separately:
  • \(-5 \cdot 2a\)
  • \(-5 \cdot 3b\)
This property helps us break down complex expressions into simpler parts, making the math more manageable.
Simplifying Expressions
Simplifying expressions means breaking them down into their simplest form without changing their values. Using the distributive law helps us achieve this by expanding and combining like terms.
In our example with the expression \(-5(2a + 3b)\), the first step is to distribute \(-5\) to each term within the parentheses:
  • \
Multiplication of Terms
When multiplying terms in algebra, each component must be multiplied accordingly. This includes constants, variables, and their coefficients. For example, from the expression \(-5(2a + 3b)\), multiplication should be performed as follows:
  • First, \(-5\) is multiplied by \(2a\), resulting in \(-10a\).
  • Then, \(-5\) is multiplied by \ (\

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Most popular questions from this chapter

Use set-builder notation to write each set. The set of all even numbers between 9 and 99

Classify each statement as either true or false. The following sets are used: \(\mathrm{N}=\) the set of natural numbers; \(W=\) the set of whole numbers; \(\mathbb{Z}=\) the set of integers; \(\mathbb{C}=\) the set of rational numbers; \(\mathrm{H}=\) the set of irrational numbers; \(\mathrm{R}=\) the set of real numbers. $$ \mathrm{N} \subseteq W $$

Tico's scores on four tests are \(83,91\) \(78,\) and \(81 .\) How many points above his current average must Tico score on the next test in order to raise his average 2 points?

Use set-builder notation to write each set. \(\\{0,1,2,3,4\\}\)

To the student and the instructor: Writing exercises, denoted by \([\%,\) are meant to be answered using sentences. Because answers to many writing exercises will vary, solutions are not listed at the back of the book. What is the difference between rational numbers and integers?
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