Problem 50 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ -(9 r+5) $$

Step-by-Step Solution

Answer

The simplified expression is \(-9r - 5\).

Step 1: Apply the Distributive Property

To eliminate the parentheses, apply the distributive property by multiplying each term inside the parentheses by the negative sign outside the parentheses. This means you distribute the negative sign across the terms in the expression \[-(9r + 5)\]This gives us:\[-1 \cdot 9r + (-1) \cdot 5\]

Step 2: Simplify Each Term

Simplify the expression by multiplying each term:\[-9r + (-5)\]By simplifying, we have:\[-9r - 5\]

Step 3: Final Simplified Expression

There are no further simplifications necessary because the expression is already in its simplest form. The expression with no parentheses and simplified is:\[-9r - 5\]

Key Concepts

Understanding Algebra

Algebra is a fascinating world within mathematics where we use letters and symbols to represent numbers and quantities in equations and expressions. It acts as a bridge between arithmetic operations and more abstract mathematical thinking. This allows us to solve problems in a more general form.
In our exercise, we deal with an algebraic expression -(9r + 5). Here, 'r' is a variable, which is a symbol for a number that we don't yet know. Variables like 'r' can take on different values, allowing you to handle both specific and general cases in one fell swoop.
Understanding how to manipulate these algebraic expressions is key to unlocking an array of mathematical mysteries. It's like learning a language: once you're fluent in the grammar rules of algebra, you can start solving problems that would otherwise remain inaccessible to pure numerical calculations.

Simplifying Expressions

Simplifying expressions is all about making them easier to work with by applying certain rules or properties of mathematics. This might involve removing parentheses, combining like terms, or reducing fractions. In our particular exercise, we use the distributive property to simplify -(9r + 5).
The distributive property in math allows you to multiply a single term across multiple terms enclosed in parentheses. When you see an expression like -(9r + 5), you can think of the negative sign as a -1 multiplied by the term inside the parentheses. So, you distribute the -1 to each term, which results in:

• -1 multiplied by 9r, resulting in -9r
• -1 multiplied by 5, giving you -5

Once these individual products are found, we can put them together to rewrite the expression without parentheses: -9r - 5. This process makes it more manageable, especially when working with larger or more complex equations.

Handling Negative Numbers

Negative numbers can seem a bit tricky, but they're just as important as positive numbers. They're used to represent values less than zero, and are essential in various real-world contexts, like debts or temperatures below freezing.
In our exercise, the distributive property involves a negative multiplier. This means that every term within the parentheses is affected by this negative sign. When you multiply a negative number by a positive number, you get a negative result.
For example, multiplying -1 by 9r gives us -9r. Similarly, multiplying -1 by 5 results in -5. When handling negative numbers, always remember:

• Multiplying or dividing two negative numbers yields a positive result
• Multiplying or dividing a negative and a positive number yields a negative result
• When adding a negative number, it's like subtracting a positive one

Understanding these rules helps you simplify expressions more effectively and navigate through negative numbers with confidence.