Problem 35 - Prealgebra

Question

Apply the distributive property to each expression and then simplify. $$5(2 x-3)+4$$

Step-by-Step Solution

Answer

The simplified expression is \(10x - 11\).

Step 1: Apply the Distributive Property

Use the distributive property to multiply each term inside the parentheses by the factor outside. The expression is \(5(2x - 3)\), so distribute the \(5\) to both \(2x\) and \(-3\). Therefore, \(5 \cdot 2x = 10x\) and \(5 \cdot (-3) = -15\). Write the expression as \(10x - 15\).

Step 2: Combine Like Terms

After distributing, the expression is \(10x - 15 + 4\). Combine the constant terms \(-15\) and \(+4\), which gives \(-15 + 4 = -11\). The expression becomes \(10x - 11\).

Key Concepts

Like Terms

Understanding the concept of 'like terms' is crucial in algebra. Like terms are terms that have the same variables raised to the same powers. They are gathered together because they behave similarly during arithmetic operations like addition or subtraction.
Here are a few tips to identify like terms:

• Check if the terms share the same variable (e.g., all terms with the variable \(x\)).
• Ensure the variables are raised to the same power (e.g., \(x^2\) with \(x^2\), but not \(x\) with \(x^2\)).

When simplifying algebraic expressions, combining like terms involves adding or subtracting the coefficients (the numerical part) of the terms, while keeping the variable part unchanged. For example, in the expression \(3x + 2x\), you add the coefficients \(3\) and \(2\) to get \(5x\).
Understanding and mastering this concept will help you in simplifying expressions and solving algebraic equations.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. This makes them easier to work with and understand. The process often involves applying mathematical operations, such as distributing factors and combining like terms.
The key steps in simplifying expressions typically are:

• Use the distributive property to eliminate parentheses.
• Combine like terms by adding or subtracting their coefficients.

For the given problem, the expression \(5(2x - 3) + 4\) was simplified by first using the distributive property: multiplying \(5\) by each term inside the parentheses, resulting in \(10x - 15\). Then, by combining the like terms, specifically the constants \(-15\) and \(+4\), it further simplifies to \(10x - 11\).
Simplifying expressions not only makes calculations easier but also helps in understanding the structure and relationships within mathematical problems.

Prealgebra

Prealgebra serves as the foundation for all higher-level math courses. It's where students begin to use algebraic techniques to solve problems. In prealgebra, you will encounter concepts like the distributive property, combining like terms, and simplifying expressions, which are essential building blocks for more complex algebra. Key aspects of prealgebra include:

• Understanding basic mathematical operations and their properties, like addition, subtraction, multiplication, and division.
• Introducing variables as placeholders for numbers and learning how to manipulate expressions containing them.
• Emphasizing the importance of following mathematical procedures and rules to solve problems correctly.

By grasping prealgebra concepts, students gain confidence and develop problem-solving skills that are vital for tackling more advanced topics. As seen in the exercise, using the distributive property and combining like terms are fundamental skills that students begin to master during prealgebra, setting the stage for success in future algebra courses.