Problem 14 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(10+5(2 y+7)\)

Step-by-Step Solution

Answer

10y + 45

Step 1: Distribute the 5

Multiply the term 5 by each term inside the parentheses: 5 * 2y + 5 * 7

Step 2: Write the result of distribution

After distributing the 5, the expression becomes: 10 + 10y + 35

Step 3: Combine like terms

Combine the constants 10 and 35 to simplify the expression: 10y + 45

Key Concepts

Distribution in Algebra

Distribution is a crucial method in algebra that helps simplify expressions. It's used to multiply a single term by each term within parentheses. For example, when faced with the expression \(5(2y + 7)\), distribution requires multiplying the 5 by both \(2y\) and 7.

This process of distribution follows these steps:

• Multiply the constant outside the parentheses with each term inside.
• Combine the products.

In our case, \(5 * 2y = 10y\) and \(5 * 7 = 35\). After applying distribution, the expression \(5(2y + 7)\) becomes \(10y + 35\).

Distribution simplifies more complex problems and is extensively used in solving equations, making it an invaluable algebraic tool.

Combining Like Terms

Once you've distributed terms in an algebraic expression, the next step often involves combining like terms. Like terms are terms that have the same variable raised to the same power. Combining them simplifies the expression further.

Similar to sorting laundry, terms with the same 'label' (variable) get combined. For example, in the expression \(10 + 10y + 35\), 10 and 35 are constants and can be added together. This yields \(45\).

The expression then reduces further to \(10y + 45\). Remember to only combine terms with identical variables (or no variables). This method ensures simplified and manageable expressions. This can often make solving equations more straightforward.

Algebraic Operations

Simplifying algebraic expressions involves several operations like addition, subtraction, multiplication, and division. Let's break down the process with our example:

• Single out each operation: Distribute, combine like terms, and simplify.
• After distributing in \(5(2y + 7)\), you get \(10y + 35\).
• Adding the constant 10 results in \(10 + 10y + 35\).

Next, apply addition or subtraction to combine like terms, simplifying further. Here, adding 10 and 35 gives \(45\).

The result is the simplified expression \(10y + 45\). By following these algebraic operations in sequence, complex expressions can become more manageable. Each operation builds on the previous, making it essential to master these basic skills.