Problem 84 - Introductory Algebra for College Students

Question

In Exercises \(81-88,\) simplify each algebraic expression by removing parentheses and brackets. $$5[2-(y+3)]$$

Step-by-Step Solution

Answer

-5y - 5

Step 1: Apply the Negative Sign to the Inner Parenthesis

The first step is to distribute the negative sign inside the inner parenthesis, so the equation becomes \(5[2 - y - 3]\).

Step 2: Simplify Inside the Bracket

Next, we combine the constants within the brackets, getting \(5[-y - 1]\).

Step 3: Apply the 5 to Everything Within the Bracket

Finally, distribute the 5 across the terms inside the brackets, yielding \(5(-y) - 5(1)\).

Key Concepts

Explaining the Distributive Property

The distributive property is a critical concept in algebra that allows you to simplify expressions by expanding multiplication across terms within parentheses. For example, when you encounter an expression such as

\( a(b + c) \),

the distributive property enables you to multiply the term outside the parentheses, in this case 'a', by each term inside the parentheses individually. This results in

\( ab + ac \).

In the exercise \( 5[2 - (y + 3)] \), we use the distributive property in the last step. Each term within the brackets is multiplied by the number 5 outside:

\( 5 \times -y \) and \( 5 \times -1 \).

This gives us the final simplified expression \( -5y - 5 \). The distributive property is essential for removing parentheses and simplifying expressions, as demonstrated in this example.

Combining Like Terms

Combining like terms is the process of simplifying algebraic expressions by adding or subtracting terms that have the same variable raised to the same power. This simplifies the expression to a form that is easier to work with or solve. For instance, in an expression such as

\( 3x + 4x \),

since both terms involve the same variable 'x' to the same power, they can be combined to result in

\( 7x \).

In the given exercise, after we applied the distributive property, we didn't have any like terms to combine within the brackets. However, understanding how to identify and combine like terms is important when dealing with more complex expressions where this step is necessary.

Eliminating Parentheses in Algebra

The process of eliminating parentheses is fundamental in simplifying algebraic expressions. It involves performing operations to remove parentheses and combine terms into a simplified form. Generally, you want to start by distributing any coefficients or negative signs across the terms within the parentheses.

Take the initial expression from our exercise \( 5[2 - (y + 3)] \). We first needed to address the innermost parentheses by distributing the negative sign, turning \( y + 3 \) into \( -y - 3 \). This was our Step 1. Next, we simplified the expression within the brackets, leading us to Step 2, which gave us \( -y - 1 \). Finally, by applying the distributive property in Step 3, we distributed the 5 across the terms inside the brackets to eliminate them, resulting in our simplified expression \( -5y - 5 \). At each stage, the goal is to systematically remove the parentheses to steadily reveal the simplest form of the expression.