Problem 70 - Algebra A Combined Function

Question

Perform each indicated operation and simplify. $$ \frac{-2 y}{-2} $$

Step-by-Step Solution

Answer

The expression simplifies to \( y \).

Step 1: Cancel Out the Negative Sign

Notice that both the numerator and the denominator have a \(-2\). When both the numerator and the denominator of a fraction are negative, you can divide them, effectively canceling the negative signs. This yields: \[ \frac{-2y}{-2} = \frac{2y}{2} \]

Step 2: Simplify the Fraction

The expression \(\frac{2y}{2}\) can be simplified. Divide the numerator and the denominator by their greatest common divisor, which is 2. This provides:\[ \frac{2y}{2} = y \]

Step 3: Conclusion: Simplified Expression

The expression \(-2y \over -2\) simplifies directly to \y\. The final answer is simply \y\.

Key Concepts

Numerator and Denominator

In any fraction, understanding the roles of the numerator and the denominator is crucial. The numerator is the top part of a fraction, representing how many parts we have. In the example \(-2y\), it tells us we have \-2y\ parts. Meanwhile, the denominator is the bottom part, showing how many equal parts the whole is divided into. Here, the denominator is \-2\, suggesting our whole is divided into negative two equal parts.
One important thing to note is that when both numerator and denominator share a negative sign, they cancel each other out. This is because dividing two negative numbers results in a positive number. Therefore, \(-2y\) divided by \-2\ simplifies to \2y/2\.
By grasping these roles, simplifying fractions becomes more understandable as you learn to manipulate and adjust both parts of the fraction effectively.

Greatest Common Divisor

The greatest common divisor (GCD) plays a key role in simplifying fractions. The GCD is the largest number that can divide both the numerator and denominator without leaving a remainder. In our example, \2y/2\, the GCD is 2.
Finding the GCD simplifies the fraction by reducing it to its simplest form. For instance:

• Identify the GCD of the numerator and denominator. Here, it's 2.
• Divide both the numerator \(2y\) and the denominator \(2\) by the GCD, 2.
• This gives us \(y\).

Effectively finding and using the GCD ensures that the fraction cannot be simplified further, leading to a clean and concise expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can be puzzling at first, but they are just a way of representing numbers in a more general form. Take the expression \2y\ as an example.

• \(2y\) consists of the constant \(2\) and the variable \(y\).
• The variable \(y\) can represent any number, which makes the expression versatile.

In algebra, simplifying means reducing the complexity of expressions or equations while maintaining their equivalence. In this context, \2y/2\ simplifies to \(y\) by cancelling out terms or combining like parts.
Getting comfortable with algebraic expressions enables you to handle more complicated problems and apply these skills to a variety of mathematical settings.