Problem 50 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{2}{y}+z $$

Step-by-Step Solution

Answer

The expression \( \frac{2}{y} + z \) cannot be simplified further; it's already in simplest form.

Step 1: Understand the Expression

The given expression is \( \frac{2}{y} + z \). It consists of two terms: a fraction \( \frac{2}{y} \) and a variable \( z \). There are no operations between them other than addition.

Step 2: Identify Like Terms

In the expression \( \frac{2}{y} + z \), the terms \( \frac{2}{y} \) and \( z \) are not like terms. Like terms have the same variables raised to the same power. Here, one is a fraction dependent on \( y \), while the other is an independent variable \( z \).

Step 3: Simplifying the Expression

Since the terms \( \frac{2}{y} \) and \( z \) are not like terms, they cannot be simplified further by combining. The expression is already in its simplest form.

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers: the numerator and the denominator. The number above the line is called the numerator, and it specifies how many parts of the whole are being considered. The number below the line is called the denominator, and it indicates the total number of equal parts the whole is divided into. For example, in the fraction \(\frac{2}{y}\), 2 is the numerator, and \(y\) acts as the symbolic denominator.

When working with fractions, it's crucial to remember:

• If the denominator is zero, the fraction is undefined because division by zero is not possible.
• Fractions can be simplified by dividing the numerator and denominator by their greatest common divisor, provided both are numeric.
• When performing arithmetic with fractions, ensure the denominators are the same, or convert them so they are. This step is vital for addition and subtraction.

Role of Variables in Algebra

Variables are symbols used to represent unknown numbers or quantities. They allow expressions and equations to be written in a general form, making them versatile and applicable to many situations. In the expression \( \frac{2}{y} + z \), both \(y\) and \(z\) are variables.

Here are some key points about variables:

• Variables can represent any number, and this flexibility is what makes algebra powerful.
• The operation performed often depends on the variable's value, which might change depending on the situation.
• Variables are essential for creating formulas and models in mathematics and many other fields such as physics and economics.

Understanding how to work with variables means understanding their interactions in expressions and equations. It's important to note that variables raised to different powers, or variables that are not the same, are generally not like terms and cannot be directly combined.

Understanding Like Terms

In algebra, like terms are terms that have the same variable part. This means they have identical variable symbols raised to the same power. To simplify expressions, you often combine like terms. However, not all terms can be combined.

Consider the terms in our original expression: \( \frac{2}{y} \) and \( z \). These are not like terms because:

• \( \frac{2}{y} \) is a fraction involving the variable \(y\), while \(z\) is a standalone variable.
• They do not share the same variable, nor are they raised to the same power.

It's important to be able to identify like terms when simplifying expressions. Here are some tips:

• Ensure that both the variable and the exponent (if present) are exactly the same when identifying like terms.
• Only the numerical coefficients of like terms can be added or subtracted, while their variable component remains unchanged.
• Recognizing unlike terms is equally essential, as they cannot be merged; this keeps expressions in their simplest forms.