Problem 33 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(2x - 2y\).

Step 1: Identify the Expression

The expression given is \(2(x-y)\). We need to apply the distributive property to this expression, which involves multiplying the term outside the parenthesis by each term inside the parenthesis.

Step 2: Apply the Distributive Property

According to the distributive property: \(a(b+c) = ab + ac\). Here \(a = 2\), \(b = x\), and \(c = -y\). Applying this, we get: \(2(x) + 2(-y)\).

Step 3: Perform the Multiplications

Multiply each term: \(2 \cdot x = 2x\) and \(2 \cdot (-y) = -2y\). Therefore, the expression becomes \(2x - 2y\).

Step 4: Simplify the Expression

There is no further simplification needed for the expression \(2x - 2y\) as there are no like terms to combine.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In the expression \(2(x-y)\), the number 2 is the constant or coefficient, while \(x\) and \(y\) are the variables.
The role of algebraic expressions is to represent mathematical relationships in a more abstract form. This allows for general solutions rather than specific numerical ones.
Unlike equations, which have an equals sign, expressions do not equate to anything; they are simply a combination of terms. Understanding algebraic expressions is crucial for solving mathematical problems efficiently.

Simplifying Expressions

Simplifying expressions involves applying mathematical rules to reduce an expression to its simplest form. This could mean combining like terms or using properties such as the distributive property.
When we simplify \(2(x-y)\) using the distributive property, we multiply each term inside the parentheses by the term outside, giving us \(2x - 2y\).
In this case, the simplified expression is already clean because no further like terms can be combined.

• Apply properties: Use the distributive property and other algebraic rules.
• Combine like terms: Terms that have the same variable and power.

Simplifying expressions makes them easier to understand and work with and is a fundamental skill in algebra.

Prealgebra Concepts

Prealgebra sets the stage for all future algebraic learning. It covers basic arithmetic to more complex topics like fractions, decimals, and basic properties.
The distributive property is a critical prealgebra concept. It shows how multiplication interacts with addition or subtraction within an expression, giving \(a(b + c) = ab + ac\).

• Understand basics: Grasp simple operations and how they extend to variables.
• Properties of operations: Learn about commutative, associative, and distributive properties.

By mastering these prealgebra concepts, students can confidently approach more advanced topics in later studies. Concepts like the distributive property are not just rules to memorize. They are tools that assist in solving and simplifying expressions.