Problem 41 - Algebra 2

Question

Simplify each expression. $$ \frac{2 x}{3 y}+5 $$

Step-by-Step Solution

Answer

The expression is already simplified as \( \frac{2x}{3y} + 5 \).

Step 1: Understand the Expression

We are given the algebraic expression \( \frac{2x}{3y} + 5 \). Our task is to simplify it.

Step 2: Check for Common Terms

Examine the expression \( \frac{2x}{3y} + 5 \) to see if there are any like terms that can be combined. In this expression, \( \frac{2x}{3y} \) and \( 5 \) are separate terms and cannot be combined because they do not share a common variable base.

Step 3: Simplified Form

Since \( \frac{2x}{3y} \) and \( 5 \) do not share any like terms, the expression \( \frac{2x}{3y} + 5 \) is already in its simplest form.

Key Concepts

Simplification of Expressions

Simplifying an algebraic expression means to make it as concise and straightforward as possible. The goal is to break down complex expressions into their simplest form without altering their value. This often involves combining like terms, reducing fractions, and applying mathematical operations.

In our exercise, the expression \( \frac{2x}{3y} + 5 \) needs to be checked for any potential simplifications. It's essential first to identify whether any terms can be combined or simplified further. If they can't be, as in this case, the expression is already in its simplest form. Simplification is crucial as it makes equations easier to solve and understand.

When approaching such tasks, always:

• Look for common factors or terms.
• Rearrange terms to bring any like terms together.
• Perform basic arithmetic operations like addition and reduction whenever possible.

Like Terms

Like terms are those terms in an algebraic expression that have the same variable raised to the same power. Identifying and combining like terms is a key step in simplifying expressions.

In the expression \( \frac{2x}{3y} + 5 \), we see two distinct terms. \( \frac{2x}{3y} \) is a fraction involving variables \( x \) and \( y \), while \( 5 \) is a constant term. Because these terms do not share the same variable or structure, they cannot be combined.

To become proficient in working with like terms, remember:

• Always compare the variables and their exponents.
• Only combine terms that match completely.

Understanding this will help keep your expressions in the simplest form without making errors.

Variables and Constants

Variables and constants are foundational elements of algebraic expressions. A variable represents an unknown or changeable number, often denoted by letters such as \( x \), \( y \), etc. In our expression \( \frac{2x}{3y} + 5 \), \( x \) and \( y \) are variables which imply that their values can change, impacting the overall expression.

A constant, on the other hand, is a fixed number that does not change. In this case, \( 5 \) is a constant, meaning its value is invariant in the expression.

When working with algebra:

• Identify variables and constants clearly.
• Treat constants as terms that offer no variability – they remain the same no matter what the variable values are.
• Understand that variables in fractions or equations may require additional steps to isolate or solve.

Grasping these differences helps in correctly simplifying and solving algebraic expressions.