Problem 108 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 10-5(-5 g-1) $$

Step-by-Step Solution

Answer

The simplified expression is \(25g + 15\).

Step 1: Distribute the Negative Sign

The expression given is \(10 - 5(-5g - 1)\). Begin by distributing the \(-5\) across the parentheses. This means you will multiply \(-5\) by each term inside the parentheses:\(-5(-5g) = 25g\)\(-5(-1) = 5\).The expression becomes:\(10 + 25g + 5\).

Step 2: Combine Like Terms

Now that we have transformed the expression to \(10 + 25g + 5\), we need to combine like terms. The terms \(10\) and \(5\) are constants and can be combined:\(10 + 5 = 15\).This simplifies our expression to:\(25g + 15\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to break down expressions to make them easier to handle. It involves multiplying a term outside of parentheses by each term inside the parentheses.

For example, given the expression \(a(b + c)\), the distributive property states that this can be rewritten as \(ab + ac\). Here’s how you apply it:

• Multiply the term outside the parentheses by the first term inside.
• Multiply the term outside the parentheses by the second term inside.
• Add or subtract the results as indicated in the original expression.

This property is especially useful when dealing with negative numbers. In the expression \(10 - 5(-5g - 1)\), the \(-5\) is distributed across \(-5g - 1\). This ensures each component inside the parentheses is effectively multiplied by \(-5\), resulting in a positive product in this case.

Understanding how to use the distributive property lets you simplify complex algebraic expressions systematically. It transforms expressions into a form that highlights like terms, which is critical for further simplification.

Combining Like Terms

Once you have used the distributive property, the next step is combining like terms. Like terms are terms in an expression that have identical variable parts raised to the same power. Only these terms can be combined because they represent the same type of quantity.

For instance, in the expression \(10 + 25g + 5\), we look for terms that can be added together. Here:

• The constants \(10\) and \(5\) are like terms because they do not have a variable part. They combine to \(15\).
• The term \(25g\) cannot be combined with the constants because it has the variable \(g\).

So, combining like terms involves identifying terms that you can add together. After using the distributive property, this step simplifies the expression further, making it more manageable for interpretations or further operations.

Keep in mind, always keep track of signs (positive or negative) preceding the terms, as they affect how you combine them.

Simplifying Expressions

Simplifying expressions is all about making an algebraic expression as straightforward and concise as possible. It's a process that usually combines several algebraic techniques, including using the distributive property and combining like terms.

The goal is to reduce the expression to its simplest form where you can't simplify any further. In our example, after applying the distributive property and combining like terms, we transformed the original expression:

• Started with \(10 - 5(-5g - 1)\).
• Simplified to \(10 + 25g + 5\) with the distributive property.
• Finally reduced it to \(25g + 15\) by combining like terms.

Each simplification step is crucial to ensuring clarity and ease of understanding whenever you're solving or interpreting algebraic expressions.

Mastering the skills of simplifying expressions empowers you to solve more complex equations and enhance your mathematical problem-solving capabilities. Embrace each step and practice consistently to strengthen your algebraic fluency.