Problem 42 - Elementary Algebra

Question

Solve. $$ 3(5 x+4)+5 x-8 $$

Step-by-Step Solution

Answer

The expression simplifies to \( 20x + 4 \).

Step 1: Distribute the 3

First, distribute the 3 across the terms inside the parentheses. This results in: \( 3 imes 5x + 3 imes 4 = 15x + 12 \).So the expression becomes: \( 15x + 12 + 5x - 8 \).

Step 2: Combine Like Terms

Next, combine the like terms in the expression: Combine the terms that have \(x\): \( 15x + 5x = 20x \).Combine the constant terms: \( 12 - 8 = 4 \).So, the expression simplifies to: \( 20x + 4 \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions, especially when dealing with parentheses. It lets us multiply a single term by each term within the parenthesis. In this exercise, the distributive property is applied to the expression \(3(5x + 4)\). This means we multiply the 3 by each component within the parentheses:

• First, multiply 3 by \(5x\), which results in \(15x\).
• Then, multiply 3 by 4, which gives us 12.

Bringing these together, the distributed expression becomes \(15x + 12\). This is how we use the distributive property to break down complex expressions into simpler parts. The property is mathematically expressed as \(a(b + c) = ab + ac\), showcasing how a single factor outside the parentheses gets applied to each term inside.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. It enables us to reduce expressions by grouping similar components. In the example \(15x + 12 + 5x - 8\), we need to identify and combine the terms that belong together:

• First, look at the terms containing \(x\). Here, they are \(15x\) and \(5x\). By combining them, we add the coefficients: \(15 + 5\) to get \(20x\).
• Next, focus on the constant numbers, which are 12 and -8. By subtracting 8 from 12, we get 4.

Combining like terms helps to simplify the expression by reducing it to fewer components. The goal is to have each type of term appear just once in the final expression, like \(20x + 4\). This step ensures that all similar terms are neatly packaged together, making the expression easier to understand and use.

Simplifying Expressions

Simplifying an expression involves performing all possible operations and making the expression as concise as possible. After distributing terms and combining like terms, you often arrive at a much more straightforward version of the original expression. In this exercise, we started with:\(3(5x + 4) + 5x - 8\),and transformed it into:\(20x + 4\).The simplification process included using the distributive property to remove parentheses and then combining like terms to reduce the expression. Simplification makes expressions easier to work with, providing clarity and ease in mathematical calculations or in solving further equations. Maintaining a simple form also minimizes errors in arithmetic operations or when plugging expressions into formulas. The end result is the most efficient and accessible version of the given problem.