Problem 81 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -3(2 x-5) $$

Step-by-Step Solution

Answer

-6x + 15

Step 1: Identify the Distributive Property

The distributive property states that for any numbers, a, b, and c, the expression: \[ a(b + c) = ab + ac \] or \[ a(b - c) = ab - ac \]

Step 2: Apply the Distributive Property

In this problem, the expression is \(-3(2x - 5)\). Here, \(a = -3\), \(b = 2x\), and \(c = 5\). Applying the distributive property gives: \[ -3(2x - 5) = -3 \times 2x - (-3) \times 5 \]

Step 3: Simplify the Expression

Multiply the terms: \[ -3 \times 2x = -6x \] and \[ -(-3) \times 5 = 15 \] Therefore, the expression simplifies to: \[ -3(2x - 5) = -6x + 15 \]

Key Concepts

algebraic expressions

An algebraic expression includes numbers, variables, and arithmetic operations combined together. Variables are symbols that represent unknown values, typically shown as letters like x, y, or z. These expressions help us describe relationships and solve various problems in algebra.
Examples of algebraic expressions are:

• 3x + 2
• 7y - 4
• 2a^2 + 3a - 1

In the exercise given,
we need to understand each part of the expression. For instance, -3(2x - 5) includes:

• Coefficient: -3
• Variable: x
• Constant: -5

Understanding how to manipulate and simplify these expressions is key to solving algebraic problems.

simplifying expressions

Simplifying expressions means making them easier to work with by combining like terms, using properties such as the distributive property, and performing arithmetic operations. The goal is to rewrite the expression in its most simplified form.
Consider the expression we are given:

• -3(2x - 5)

By applying the distributive property, we can break down the expression into simpler parts:

• First, distribute -3 to both 2x and -5
• Then, perform the multiplication
• Combine the results to create a new, simpler expression

The result is:
-3(2x - 5) = -6x + 15
Steps to simplify:

• Distribute: -3 * 2x = -6x
• Distribute: -3 * (-5) = 15
• Combine: -6x + 15

Through simplification, the expression is now easier to interpret and use in further calculations.

multiplication in algebra

Multiplication in algebra extends the basic concept of multiplication to include variables and constants. Multiplication can be between:

• Two constants: 3 * 2 = 6
• Constant and variable: 3 * x = 3x
• Two variables: x * y = xy

In our exercise, we've applied multiplication within the distributive property. Here's a closer look:
Given: -3(2x - 5)

• Multiply -3 by 2x: -3 * 2x = -6x
• Multiply -3 by -5: -3 * (-5) = 15

This shows how multiplication distributes over addition and subtraction. The distributive property states that:
a(b + c) = ab + ac
In essence, we multiply each term inside the parentheses by the term outside. Understanding and mastering multiplication with variables helps you tackle more complex algebraic expressions.