Problem 30 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$2(x-3)$$

Step-by-Step Solution

Answer

The expression without parentheses is \(2x - 6\).

Step 1: Apply the distributive property

The distributive property is used to remove the parentheses in this expression by multiplying the '2' outside the parentheses with every term inside the parentheses. We get \(2*x - 2*3\).

Step 2: Perform the operations

Now we can calculate the multiplication operations: \(2*x - 2*3 = 2x - 6\)

Key Concepts

Algebraic Expressions

Algebraic expressions are like math sentences that involve numbers, variables (like \( x \)), and operations (like addition or subtraction). These expressions are a way to represent real-world problems and mathematical ideas in a concise form. The equation given, \( 2(x - 3) \), is an example of an algebraic expression. Here, it's expressed with a coefficient (the number 2), a variable \( x \), and a constant (the number 3) as part of the expression.

• Variable: Represents an unknown value and is usually denoted by letters like \( x \), \( y \), etc.
• Coefficient: A number placed before a variable indicating multiplication (In this case, 2 is the coefficient of \( x \)).
• Constant: A fixed number that does not change (here, it's 3).

By understanding these components, we can approach problems logically and organize complex equations into simpler parts.

Multiplication

Multiplication in algebra is a way to quickly add a number to itself a set number of times. In algebraic expressions like \( 2(x-3) \), multiplication is not only about numbers. It also involves multiplying through different components of expressions such as variables and constants.
When practicing multiplication in algebra, you might encounter situations where a coefficient is multiplied with both a variable and a constant, which is the essence of the distributive property. Using multiplication, the term "\( 2(x - 3) \)" expands to "\( 2*x - 2*3 \)." This highlights:

• Combining multiplication with the distributive property simplifies expressions without changing their value.
• Ensuring each term inside the parentheses interacts with the constant outside, leading to a correct expansion.

Practicing these concepts keeps equations manageable and strengthens your understanding.

Parentheses Removal

Parentheses in mathematical expressions indicate which operations to perform first. In algebra, parentheses can often be seen with operations like addition, subtraction, or multiplication. Removing them requires applying the distributive property, as shown in the expression \( 2(x - 3) \).
Removing parentheses involves redistributing the terms outside with those inside, leading to a simpler expression. Here's the process we applied:

• The number outside the parentheses (in this case, 2) is multiplied by each term within the parentheses.
• This expansion removes parentheses by dispersing the outside number to each of the inside components, resulting in: "\(2*x - 2*3\)" which further simplifies to "\(2x - 6\)."

Understanding how to correctly handle parentheses in algebraic expressions enhances problem-solving skills and ensures seamless manipulation of expressions.