Problem 41 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(2(6 x-3 y)\)

Step-by-Step Solution

Answer

The simplified expression is \(12x - 6y\).

Step 1: Apply the Distributive Property

The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the results. For the expression \(2(6x - 3y)\), distribute the \(2\) to both terms inside the parentheses: \(2 \times 6x\) and \(2 \times -3y\).

Step 2: Simplify Each Term

Calculate each term from the previous step. For \(2 \times 6x\), the result is \(12x\). For \(2 \times -3y\), the result is \(-6y\).

Step 3: Combine Results

After distributing and simplifying each term, combine them to express the final simplified expression. Therefore, the expression becomes \(12x - 6y\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases combining numbers, variables, and operations. They allow mathematicians to represent problems symbolically. In simpler terms, an algebraic expression is built using various mathematical operations such as addition, subtraction, multiplication, and division, while also involving variables represented by letters like \(x\) and \(y\). One key aspect to remember is that variables can take on different numerical values. This characteristic makes algebraic expressions extremely useful in modeling numerous real-world situations. For instance, \(2(6x - 3y)\) is an algebraic expression where \(6x\) and \(-3y\) are terms connected by subtraction and multiplied by 2 outside the parentheses. It is critical to understand that simplifying such expressions often involves the application of rules or properties such as the distributive property.

Simplification

Simplification involves reducing an expression to its most compact and easily interpretable form without changing its value. When simplifying expressions such as \(2(6x - 3y)\), the first step is often distribution. This is applying the distributive property, where you multiply the number outside the parentheses by each term within the parentheses. Here, distribute \(2\) to both \(6x\) and \(-3y\), resulting in \(12x\) and \(-6y\) respectively. After distribution, it is essential to combine like terms, if any, to further simplify the expression. In our case, there are no like terms to combine, thus simplifying the expression results in \(12x - 6y\). Always aim to present your final answer in the simplest form, which makes further mathematical operations or interpretations much more accessible.

Prealgebra Concepts

Prealgebra concepts serve as the foundation for all subsequent mathematical learning. These concepts focus on arithmetic operations combined with basic algebraic skills. The primary goal at this level is to transition smoothly from arithmetic to algebra. A crucial prealgebra concept is understanding how properties such as the distributive property work. The distributive property states that multiplying a whole expression by a number is the same as multiplying each part of the expression individually by that number. For example, in \(2(6x - 3y)\), the 2 multiplies each term within the brackets, illustrating this principle.
This understanding not only simplifies complex expressions but also enhances overall math problem-solving abilities, making advanced algebraic concepts easier to grasp in the future.