Problem 34 - Prealgebra

Question

Apply the distributive property to each expression and then simplify. $$2(5 x+1)+10$$

Step-by-Step Solution

Answer

The simplified expression is \(10x + 12\).

Step 1: Distribute the 2

Use the distributive property to multiply 2 by each term inside the parentheses. For this expression, that means multiplying 2 by both \(5x\) and \(1\). This gives us: \(2 \cdot 5x + 2 \cdot 1 = 10x + 2\).

Step 2: Add Remaining Terms

Now, we need to bring down the remaining part of the expression, which is \(+ 10\). The expression now reads: \(10x + 2 + 10\).

Step 3: Simplify the Expression

Combine like terms. In this case, combine \(2\) and \(10\) to simplify the expression to: \(10x + 12\).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making them as straightforward as possible. To do this, we perform operations in the simplest form while ensuring everything remains balanced. Here’s how it goes: When you have an expression involving addition, subtraction, multiplication, or division, the aim is to condense it to a form that's easier to interpret.

• First off, you want to deal with any parentheses using the distributive property. This breaks down tricky parts into manageable pieces.
• Then, you look for any similar terms – often called "like terms" – and combine them. This step sheds off the extra weight, leaving you with a leaner expression.
• Simplifying doesn’t change the value of the expression; it only makes reading and solving easier!

By taking complex expressions and breaking them down, you not only solve problems faster but also develop a fundamental skill necessary in higher mathematics.

Combining Like Terms

Combining like terms means simplifying an expression by merging terms that have the same variable part. These terms have identical variables raised to the same power. Think of it as gathering all the apples in one basket and all the oranges in another. Here's a closer look:

• Identify terms that are alike, meaning their variable parts must be identical. For example, in the expression \(10x + 2 + 10\), the terms \(2\) and \(10\) are like terms because they are constant terms without any variables attached and can be added together.
• Once you've found these like terms, add or subtract their coefficients while keeping the variable parts unchanged.
• By doing this, you simplify your expression, making it easier to interpret and work with further. It’s particularly helpful in balancing equations or preparing them for further solving.

Combining like terms is crucial because it sets the foundation for more advanced algebraic processes and ensures that expressions are presented in the simplest form possible.

Prealgebra

Prealgebra serves as the gateway to the more complex world of algebra. It creates the groundwork by introducing fundamental concepts and operations. Think of it as building the foundation of a house - without a strong base, the structure cannot stand. Here’s why it matters:

• Prealgebra introduces basic operations such as addition, subtraction, multiplication, and division with whole numbers, fractions, and decimals. These activities are the bread and butter of mathematical operations.
• Concepts like the distributive property, discussed in the given exercise, show how multiplication interacts with addition and subtraction in more complicated expressions.
• The more comfortable you get with prealgebra, the more prepared you’ll be for advanced topics such as algebra, trigonometry, and calculus. It trains your brain to look for patterns and develop problem-solving strategies.

Mastering prealgebra provides the necessary skills to tackle all mathematical challenges ahead, placing you firmly on the path to success in math and beyond.