Problem 44 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{22 r+10}{-2} $$

Step-by-Step Solution

Answer

The simplified expression is \(-11r - 5\).

Step 1: Apply Distributive Property

Firstly, distribute the denominator -2 to \(22r\) and \(10\) individually. This results in: \(\frac{22r}{-2} + \frac{10}{-2}\)

Step 2: Perform Each Division

Perform the division for each term: \(22r\) divided by -2 is \(-11r\), and 10 divided by -2 is \(-5\). So, the simplified expression becomes: \(-11r - 5\)

Key Concepts

Distributive Property

When simplifying expressions, the distributive property is a powerful tool that helps to break down complex expressions into simpler parts. This property allows us to distribute a single common factor across terms inside brackets or a division, thereby making calculations more manageable.

In the given exercise, the expression \( \frac{22r + 10}{-2} \) involves distributing the denominator \(-2\) to each term in the numerator. It operates like spreading out the division across each term separately. Think of it as serving a slice of pizza to each member seated at a table!

By distributing, we rewrite the expression as two separate fractions: \( \frac{22r}{-2} \) and \( \frac{10}{-2} \). This action sets the stage for further simplification by making sure each term is handled on its own terms, quite literally.

Division of Terms

Dividing terms individually is a crucial step in expression simplification, especially when a number or variable is shared among terms. This particular method ensures accuracy in deriving a cleaner expression.

In our exercise, once we distributed \(-2\) across \(22r\) and \(10\), the next step was to perform the division for each separate term. Here is how it works:

• For \( \frac{22r}{-2} \), divide 22 by \(-2\) to get \(-11r\).
• For \( \frac{10}{-2} \), divide 10 by \(-2\) to end up with \(-5\).

By dividing terms carefully, we preserve mathematical integrity and pave the way towards a tidy final expression. Remembering to simplify each term reinforces not just the solution's validity, but also your understanding of how algebraic expressions behave.

Simplification Steps

Simplifying an expression involves a sequence of logical steps that reduce an equation or formula to its most concise form. Each step in the process is an essential brick in the construction of the final outcome.

The simplification of our given expression \( \frac{22r + 10}{-2} \) included several key stages:

• Use the distributive property: Break down the expression by applying \(-2\) to each part of the numerator.
• Perform division on each term: Calculate the simplified terms \(-11r\) and \(-5\).
• Combine results: The separate simplified results \(-11r\) and \(-5\) are put together to form the final expression: \(-11r - 5\).

By following these simplification steps, we achieve a streamlined expression that retains all original values in a cleaner, more workable form. Such steps render the problem-solving process systematic and reinforce the core skills of algebraic manipulation.