Problem 67 - Algebra 2

Question

PREREQUISITE SKILL. Use the Distributive Property to find each product. $$ 2(x+y) $$

Step-by-Step Solution

Answer

The expanded form is \(2x + 2y\).

Step 1: Understand the Distributive Property

The Distributive Property states that for any numbers or expressions \(a\), \(b\), and \(c\), the expression \(a(b + c)\) can be expanded to \(ab + ac\). This means that you multiply \(a\) by each term inside the parentheses and add the products.

Step 2: Identify the Elements

In the expression \(2(x+y)\), \(2\) is the number outside the parentheses, \(x\) is the first term inside, and \(y\) is the second term inside the parentheses.

Step 3: Apply the Distributive Property

Use the Distributive Property to expand \(2(x+y)\). This means multiplying \(2\) by each term inside the parentheses: first by \(x\) and then by \(y\).

Step 4: Perform the Multiplications

Calculate the products: \(2 \times x = 2x\) and \(2 \times y = 2y\).

Step 5: Write the Expanded Expression

Combine the results of the multiplications to write the expanded form: \(2x + 2y\).

Key Concepts

Expression Expansion

Expression expansion is a fundamental concept in algebra, allowing you to transform and simplify expressions by making them more understandable. In our example, we used the Distributive Property to expand the expression \(2(x+y)\).
This allows us to express the original formula in a more detailed form, showing each term separately.

• The expression \(2(x+y)\) involves a multiplication of two factors.
• The first factor is the number outside the parentheses, and the second is the sum inside the parentheses \((x + y)\).

To expand \(2(x+y)\), we proceed by distributing or multiplying the outer factor through each term inside the parentheses. This results in each term being multiplied individually by \(2\), which simplifies the expression to \(2x + 2y\). This expanded form gives a clearer view of the entire expression, which can be useful in solving equations or simplifying further.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations that represent a value or set of values. In the expression \(2(x+y)\), both \(x\) and \(y\) are variables that can take different values, and \(2\) is a constant.
Algebraic expressions allow us to model real-world situations and solve problems by representing unknown quantities.

• Variables, like \(x\) and \(y\), can stand in for numbers that might not be immediately known.
• Constants, like \(2\), are fixed values in the expression.
• Operations, such as addition or multiplication, define how the terms interact.

In our example, the expression \(2(x+y)\) represents a situation where the sum of two values is doubled. By manipulating and simplifying the expression using expansion, we can more easily solve for specific variables or compare expressions. Understanding the components of algebraic expressions is key to solving algebra problems effectively.

Mathematical Operations

Mathematical operations are the backbone of algebra and mathematics as a whole. These operations, such as addition, subtraction, multiplication, and division, are used to manipulate numbers and expressions to solve equations and find desired results. In our task, we heavily leveraged the multiplication operation as part of the Distributive Property.
When we perform mathematical operations:

• Each operation follows a specific set of rules that govern how expressions are conventionally evaluated, known as the order of operations.
• In the expression \(2(x+y)\), multiplication is the main operation used to distribute \(2\) through the terms \(x\) and \(y\).
• Mental calculations or steps carried out involve breaking down the operation, as seen in the multiplication of \(2\) with \(x\) and \(y\). This involves calculating two separate products: \(2\times x\) and \(2\times y\).

Grasping these operations ensures we can solve complex problems by treating each part of an expression according to its mathematical instruction. By mastering these basics, a student can expand, simplify, and solve not just algebraic expressions, but a wide variety of mathematical problems.