Problem 1 - Elementary and Intermediate Algebra

Question

To _____ the expression \(5(6 x)\) means to write it in simpler form: \(5(6 x)=30 x\)

Step-by-Step Solution

Answer

The simplified form is \(30x\).

Step 1: Understand the Expression

The expression given is \(5(6x)\). It consists of a number, 5, multiplied by the product of another number, 6, and a variable, \(x\).

Step 2: Apply the Distributive Property

To simplify the expression \(5(6x)\), apply the distributive property of multiplication over addition. This property allows you to multiply each term inside the parentheses by the factor outside: \(5 \times (6x) = (5 \times 6) \times x\).

Step 3: Perform the Multiplication

Calculate the multiplication of the numbers outside the variable: \(5 \times 6 = 30\).

Step 4: Write the Simplified Expression

Combine the result of the multiplication with the variable. The expression simplifies to \(30x\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is all about reducing them to their most straightforward form without changing their value. Consider the expression given in the exercise, which is \(5(6x)\). At first glance, it might seem a bit complex because it involves multiple elements like numbers and variables all tied together with multiplication.

The goal of simplification is to make such expressions more manageable or easier to work with, often by removing parentheses and performing inside operations. In this case, we took advantage of the distributive property to simplify \(5(6x)\) into \(30x\). It is vital to ensure that while simplifying, we maintain the original expression's equivalency. This simpler form typically involves fewer components, making further calculations seamless.

Here are a few tips for simplifying expressions:

• Break down each part of the expression step by step.
• Apply appropriate mathematical properties or rules.
• Keep track of the variables and coefficients (the numbers in front of the variables).

Multiplication in Algebra

In algebra, multiplication doesn't only involve numbers but also variables and potentially complex expressions. For example, in the exercise, we tackled multiplying a whole number by a term composed of another number and a variable, specifically \(5 \times (6x)\).

This process can be a bit confusing initially, but it's critical to remember that coefficients multiply together, and we extend this with the variable. When we multiply \(5\) and \(6x\), the coefficients (5 and 6) multiply to become \(30\), and the variable \(x\) remains unchanged. Hence, the multiplication process results in \(30x\).

Multiplying expressions involves:

• Identifying parts of the expression that need to be multiplied.
• Performing the multiplication straightforwardly, just as with basic arithmetic but keeping variables alongside.
• Making sure to keep the variable with the new coefficient after multiplication.

Understanding and applying these steps is pivotal to mastering basic algebra multiplication.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and mathematical operators like addition, subtraction, multiplication, and division. They form the building blocks of algebra, helping us to model real-world situations through equations.

In our exercise, the expression \(5(6x)\) is an example of an algebraic expression. It combines known quantities (5 and 6) with an unknown quantity, the variable \(x\). Variables are essential because they allow expressions to be flexible and applicable to a variety of problems where the specific values might change.

Key aspects of algebraic expressions include:

• Coefficients: The numerical parts that multiply the variables (e.g., 5 and 6 in our expression).
• Variables: Represent unknown quantities (e.g., \(x\)).
• Operations: Mathematical processes that are applied between numbers and variables (e.g., multiplication in \(5(6x)\)).

Mastering algebraic expressions involves understanding these components and knowing how to manipulate them through simplification, solving equations, or factoring—all of which are fundamental skills in algebra.