Problem 47 - College Algebra with Modeling and Visualization

Question

Apply the distributive property. $$5(y+2)$$

Step-by-Step Solution

Answer

The expression simplifies to \(5y + 10\).

Step 1: Identify the components

The expression given is \(5(y+2)\). We need to distribute the factor outside the parentheses, \(5\), to each term inside the parentheses.

Step 2: Distribute 5 to y

Multiply the factor \(5\) by \(y\), which is the first term inside the parentheses. This gives us \(5 imes y = 5y\).

Step 3: Distribute 5 to 2

Multiply the factor \(5\) by \(2\), which is the second term inside the parentheses. This results in \(5 imes 2 = 10\).

Step 4: Combine the results

Combine the results from Step 2 and Step 3 to form the expanded expression \(5y + 10\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that involve numbers, variables, and operations like addition, subtraction, multiplication, and division. They form the core of solving many mathematical problems. In the expression given, \(5(y+2)\), we have a combination of a constant term, a variable 'y', and the operations of addition and multiplication.

To break it down:

• '5' is a constant factor, representing a fixed number.
• 'y' is a variable, which indicates a number that can change or vary.
• The parentheses indicate that the addition inside should be considered as a single entity initially.

Algebraic expressions are handy because they help us form general mathematical statements. Once you understand the basic parts, solving them gets easier, especially when using properties like distribution.

The Multiplication Process in Distribution

Multiplication is a fundamental operation where we combine added groups of a number. When working with expressions like \(5(y+2)\), multiplication involves distributing the outside factor '5' into each term within the parentheses separately. This is what we call the distributive property.

The steps:
- Multiply '5' by the first term 'y': This means you calculate \(5 \times y\), which simplifies to \(5y\). The variable stays symbolically since its value is unknown.- Multiply '5' by the second term '2': You perform \(5 \times 2\) which equals \(10\). This part is numerical and can be calculated directly.

The distributive property basically transforms two simpler multiplication calculations and then combines them. Learning this property helps simplify long operations and solves expressions faster.

Simplifying Expressions Using the Distributive Property

Simplifying expressions involves making them as compact and manageable as possible. After using the distributive property on \(5(y+2)\), we end up with \(5y + 10\). But why stop at getting this expanded form?

Simplifying is about clarity and ease:

• Identify like terms: For instance, if there were another 'y' term, you could combine them.
• Ensure that you've distributed all parts correctly: By performing each multiplication step independently as shown.
• Combine results for clarity and simplicity: This means representing the expression in a form where each part is necessary and it's clearly understood.

Simplifying helps in better problem-solving by making equations more straightforward and understandable, a key skill in algebra. Once practiced, it becomes second nature to see opportunities to simplify as a part of working with expressions.