Problem 357 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ (y+4) p $$

Step-by-Step Solution

Answer

py + 4p

Step 1: Understand the Distributive Property

The Distributive Property states that for any numbers or expressions a, b, and c: \[a(b + c) = ab + ac\]In this case, we have \[(y + 4)p\].So, we will use the Distributive Property to distribute 'p' across the sum inside the parentheses.

Step 2: Distribute 'p' to each term inside the parentheses

Distribute 'p' to both 'y' and '4'. This results in:\[p \times y + p \times 4\]

Step 3: Simplify the expression

Multiply 'p' with 'y' and 'p' with '4':\[p \times y = py\]\[p \times 4 = 4p\]Combine these results to get:\[py + 4p\]

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In our exercise, the expression is \[(y+4)p\]. Here, 'y' is a variable, '4' is a constant, and 'p' is another variable. When working with algebraic expressions, we often need to perform operations like addition, subtraction, multiplication, and division. For example, the given expression involves multiplication and addition. It is important to remember that variables represent unknown values, while constants are fixed numbers. Breaking down and understanding each part of the expression is crucial for simplifying and solving algebraic problems.

Simplification

Simplification in algebra involves reducing an expression to its simplest form. The goal is to make the expression easier to work with and understand. In the case of \[(y+4)p\], simplification involves using the Distributive Property to eliminate the parentheses. To do this, we distribute 'p' to both 'y' and '4'.

• We multiply 'p' by 'y' to get 'py'.
• We multiply 'p' by '4' to get '4p'.

Combining these terms gives us the simplified expression \py + 4p\. Always ensure each step in simplification is clear and logical to avoid making errors.

Intermediate Algebra

Intermediate algebra builds on basic algebra concepts, introducing more complex operations and properties. The Distributive Property is a fundamental principle often used in intermediate algebra. It allows us to simplify expressions and solve equations more efficiently. For example, understanding that \a(b + c) = ab + ac\ helps us break down and simplify \[(y + 4)p\] into \py + 4p\. This knowledge is foundational for solving more advanced problems involving multiple variables, linear equations, and polynomials. Practicing these techniques solidifies our ability to manipulate and simplify algebraic expressions across a variety of contexts.