Problem 45 - College Algebra with Modeling and Visualization

Question

Apply the distributive property. $$-5(3 x+1)$$

Step-by-Step Solution

Answer

\(-15x - 5\)

Step 1: Identify the Distributive Property

The distributive property states that for any numbers \( a, b, \) and \( c, \) \( a(b+c) = ab + ac \). In this expression, \( a = -5, b = 3x, \) and \( c = 1 \).

Step 2: Apply the Distributive Property to First Term

Multiply \(-5\) by the first term inside the parentheses. This results in \(-5 imes 3x = -15x\).

Step 3: Apply the Distributive Property to Second Term

Next, multiply \(-5\) by the second term inside the parentheses. This results in \(-5 imes 1 = -5\).

Step 4: Combine the Results

Combine the expressions from the previous two steps. The expression becomes \(-15x - 5\).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in expressions and equations. It's like a universal language that helps in solving problems.

When dealing with algebraic expressions such as \(-5(3x+1)\), it's crucial to understand how to manipulate these expressions to simplify or solve them.
Algebra involves:

• Identifying variables and constants: In this expression, \(3x\) is a variable term while \(1\) is a constant.
• Applying mathematical operations: Here, we use distributive property to "distribute" or "multiply out" the terms within the parentheses.

Algebraic manipulation forms the foundation for solving equations and understanding more complex mathematical concepts.

Multiplication

Multiplication is one of the basic arithmetic operations where a number is added to itself a certain number of times.
In algebra, it is represented in its broader scope as well, handling products of numbers, variables or their combinations.

While handling \(-5(3x+1)\), multiplication takes on a key role, especially when applying the distributive property:

• Distribute \(-5\) to each term inside the parentheses, which results in \(-5 \times 3x = -15x\) and \(-5 \times 1 = -5\).
• Understand that multiplying with a negative number affects the sign of the result. So, both \(-15x\) and \(-5\) have negative signs due to the multiplication by \(-5\).

This process helps in rewriting expressions in simpler or more useful ways.

Linear Expressions

Linear expressions are algebraic expressions where each term is either a constant or the product of a constant and a single variable.
They are "linear" because they represent lines when graphed.
Let's dive into the linear expression \(-15x - 5\):

• Linear terms: The expression contains a linear term \(-15x\), meaning "x" is raised to the first power.
• Simplifying expressions: By distributing \(-5\) across \(3x+1\), we simplify it to \(-15x - 5\), expressing it in a linear form.

Understanding linear expressions assists in simplifying expressions and solving equations, which are useful in everyday problem-solving scenarios.