Problem 71 - Algebra A Combined Function

Question

Perform each indicated operation and simplify. $$ -5\left(-\frac{1}{5} y\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(y\).

Step 1: Distribute the Negative Sign

We need to distribute the negative sign outside the parenthesis to the fractions inside. The expression \(-5\left(-\frac{1}{5} y\right)\) becomes \(-5 \times -\frac{1}{5} \times y\).

Step 2: Multiply the Coefficients

Multiply the numbers: \(-5 \times -\frac{1}{5}\). Use the rule of signs, which states that multiplying two negative numbers gives a positive result. So \(-5 \times -\frac{1}{5} = 1\).

Step 3: Simplify the Expression

Now multiply the coefficient you found with the variable \(y\). Since we have \(1 \times y\), it simplifies to \(y\).

Key Concepts

Distributive Property

Imagine you have a stack of cards and you want to divide them into smaller stacks. The distributive property in algebra is a bit like that. It helps us break down complex multiplication problems into simpler ones. This property allows you to multiply a sum by a factor by distributing, or spreading, the multiplication over each addend. It is often represented with expressions like:

• a(b + c) = ab + ac

For the given exercise, when we see the expression \(-5\left(-\frac{1}{5} y\right)\), the negative number \(-5\) is distributed to both \(-\frac{1}{5}\) and y. This way of spreading multiplication helps to simplify the expression, making it easier to manage.

Negative Numbers

Negative numbers can sometimes feel tricky, especially since they're less familiar than positive numbers. But don’t worry, they're not as complicated as they may seem. When you're working with negative numbers in algebraic operations, there are a few key rules to remember.

• Rule of Signs: When you multiply or divide two negative numbers, the result is positive. Conversely, when you multiply or divide a negative and a positive number, the result is negative.
• Negative Times Negative: In the expression \(-5 \times -\frac{1}{5}\), both numbers are negative. By the rule of signs, multiplying them, "a negative times a negative equals a positive," results in a positive product: \(1\).

Recognizing these rules makes working with negative numbers much simpler. Even when it feels like a puzzle, just remember to apply these rules consistently.

Simplification

Simplification is the process of making an expression easier to work with. This often involves reducing expressions to their most straightforward form. In algebra, this can mean eliminating redundant operations or combining like terms to make an equation solvable or understandable.

• Basic Steps in Simplification: Look for operations like addition, subtraction, or distribution, and carry them out. In our example, after multiplying \(-5\) by \(-\frac{1}{5}\), which gives us \(1\), we then simplify the expression to just \(y\) by multiplying the variable \(y\). This reduces it to its simplest form, having effectively performed the operations.
• Importance: Simplifying expressions is crucial because it makes them easier to interpret and solve. It also removes the complexities that can obscure relationships between terms in an expression.

Thus, through simplification, we take our expression from something potentially difficult to understand and convert it into something manageable and clear.