Problem 29 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(7(x+5)\)

Step-by-Step Solution

Answer

The simplified expression is \( 7x + 35 \).

Step 1: Identify the Distributive Property

The distributive property states that for any numbers \( a \), \( b \), and \( c \), the expression \( a(b+c) \) can be expanded to \( ab + ac \). In the given expression \( 7(x+5) \), we identify \( a = 7 \), \( b = x \), and \( c = 5 \).

Step 2: Apply the Distributive Property

We distribute the 7 to both the \( x \) and the 5. This means we multiply 7 by \( x \), and 7 by 5: \( 7(x+5) = 7 \cdot x + 7 \cdot 5 \).

Step 3: Perform the Multiplication

Calculate each part of the expanded expression: \( 7 \cdot x = 7x \) and \( 7 \cdot 5 = 35 \).

Step 4: Write the Simplified Expression

Combine the results from Step 3 to write the final expanded and simplified expression: \( 7x + 35 \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that means reducing an expression to its simplest form. This process often involves combining like terms, using basic mathematical operations, and applying various algebraic properties, such as the distributive property.

• It involves breaking down complex expressions into something more manageable.
• In our exercise, after applying the distributive property, we simplify to arrive at a more straightforward expression: \(7x + 35\).

By simplifying, complex problems become easier to work with, making it a critical step in solving many algebraic equations.Remember, the goal of simplification is to make expressions easier to handle and work with by eliminating unnecessary complexity.

Mathematical Properties

Mathematical properties are the rules we follow in mathematics to solve equations and simplify expressions. The distributive property is one of these essential rules. It states that multiplying a number by a sum is the same as multiplying the number by each addend separately and then adding the products.

• The property is written as: \(a(b+c) = ab + ac\).
• This means you "distribute" the multiplication over addition inside the parentheses.

In our step-by-step solution, we applied the distributive property:

• First, multiply the 7 by each term inside the parentheses (\(x\) and \(5\)).
• This results in \(7 \cdot x + 7 \cdot 5\).
• Finally, you simplify the expression to get \(7x + 35\).

Understanding mathematical properties like the distributive property is crucial, as they provide a foundation for solving more complex algebraic problems.

Prealgebra Concepts

Prealgebra serves as the foundation for all algebraic concepts. It includes basic arithmetic, understanding symbols and terms, and properties of numbers. In our task, the distributive property is a prealgebra concept crucial for developing algebra skills.

• We learn to see how numbers and variables interact through distribution.
• This builds a foundation for more advanced topics like polynomials and quadratic equations.

In the example \(7(x+5)\), we're moving from basic arithmetic to manipulating variables. Distributing the 7 shows how one operation affects all terms inside the parentheses.This exercise exemplifies the transition in prealgebra from arithmetic math to beginning algebra, illustrating the relationship between numbers and variables clearly.