Problem 43 - Prealgebra

Question

Simplify expression. \(5(x+3)+8 x\)

Step-by-Step Solution

Answer

The simplified expression is \(13x + 15\).

Step 1: Distribute 5 into (x+3)

You begin by applying the distributive property to the expression inside the parentheses. Multiply 5 by each term inside the parentheses: \[5(x + 3) = 5 \cdot x + 5 \cdot 3 = 5x + 15\]

Step 2: Combine Like Terms

Now, add the terms with \(x\) together and the constant terms together. Initially, the expression is \[5x + 15 + 8x\]. Combine the like terms \(5x\) and \(8x\):\[(5x + 8x) + 15 = 13x + 15\]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle for understanding how to manipulate expressions. It involves distributing a multiplication over an addition or subtraction. Imagine it like sharing a factor with all the terms inside a parenthesis. If you have a number outside the parenthesis, it multiplies each term inside. In the exercise, the distributive property is applied to the expression \(5(x+3)\). Simply put:

• Multiply 5 by each term within the parentheses.
• First, multiply 5 by \(x\) leading to \(5x\).
• Then, multiply 5 by 3 to get 15.

So, distributing gives you \(5x + 15\). This step helps eliminate the parentheses and simplifies the process of further manipulation with terms.

Simplifying Expressions

Simplifying expressions is all about making the expression as straightforward as possible. This involves removing parentheses through distribution and then organizing and reducing the components of the expression. In this case, after applying the distributive property to \(5(x+3)\), you get a simpler expression: \(5x + 15\). But we're not done yet!

• After distribution, the expression becomes \(5x + 15 + 8x\).
• You need to simplify by combining terms properly, particularly focusing on like terms.

Remember, the goal is to have the expression as concise and clear as possible by minimizing the number of terms and operations.

Combining Like Terms

Combining like terms is an essential strategy that simplifies your expressions further by merging similar components. Like terms contain the same variable or variables raised to the same power. This means they can be added or subtracted from each other. In the context of our problem, you started with \(5x + 15 + 8x\).

• Identify like terms, in this case, \(5x\) and \(8x\).
• Add them together: \(5x + 8x = 13x\).
• Therefore, the expression now simplifies to \(13x + 15\).

By combining like terms, you condense the expression, making it easy to understand and work with. This step is crucial in solving algebraic equations and understanding deeper mathematical concepts.