Problem 5 - Algebra 1

Question

Use the distributive property to rewrite the expression without parentheses. $$ 5(w-8) $$

Step-by-Step Solution

Answer

The equivalent expression without parentheses is \(5w - 40\).

Step 1: Identify the terms inside the parentheses

In the expression \(5(w-8)\), the terms inside the parentheses are \(w\) and \(-8\). The number outside the parentheses is \(5\).

Step 2: Apply the distributive property

Apply the distributive property by multiplying each term inside the parentheses by the number outside the parentheses. The distributive property states that i.e. \(a(b-c) = ab - ac\). Thus, for our case, we get: \(5w - 5*8\).

Step 3: Simplify the multiplication

Carry out the multiplication operation. That is, \(5*8\) equals \(40\). This results to: \(5w - 40\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the foundation of algebra and are crucial for solving equations and other mathematical problems.
In algebra, these expressions can look like a single number, a single variable like \(x\), or a combination of numbers, variables, and arithmetic operations, for example, \(3x + 4\).
Expressions can also include parentheses, which show which operation to perform first.

• Variables are symbols that represent unknown values, usually letters like \(x, y, z\).
• Coefficients are numbers that multiply the variables, such as the \(5\) in \(5x\).
• Constants are fixed values that do not change, such as \(-40\) in the expression \(5w - 40\).

When you encounter an expression involving parentheses, like \(5(w-8)\), understanding the structure of algebraic expressions is vital to simplifying and solving them correctly.

Simplification

Simplification in algebra involves transforming an expression into its simplest form. This process makes expressions easier to work with and understand, often by combining like terms or breaking complex expressions down into smaller, manageable parts.
Using the distributive property is a key method for simplifying expressions, especially those that contain parentheses.

In the original exercise, we simplified the expression \(5(w-8)\) by:

• Distributing the \(5\) across the terms inside the parentheses, which involved multiplying \(5\) with both \(w\) and \(-8\).
• Performing the multiplication \(5 \times 8 = 40\), which led to the simplified expression \(5w - 40\).

Simplification helps to reveal the true form of an expression and makes it easier to solve equations or further transform the expression if necessary. It minimizes errors in later steps of problem-solving processes.

Multiplication

Multiplication is one of the basic arithmetic operations, used often in algebra to simplify or expand expressions. It involves calculating the product of two numbers or terms.
This operation can become complex when dealing with variables and algebraic expressions, requiring careful attention to signs and coefficients.

• In multiplication, it's important to remember the order does not affect the result: \( a\cdot b = b\cdot a \).
• In the context of the distributive property, multiplication is used to "distribute" a coefficient across terms inside parentheses.

For our example \(5(w-8)\), multiplication was carried out by:

• Multiplying \(5\) by \(w\) to get \(5w\).
• Multiplying \(5\) by \(-8\) to get \(-40\), which resulted from \(5 \times -8\).

Accurate multiplication is essential to correctly applying the distributive property, ensuring each term in the expression is correctly expanded, leading to accurate simplification.