Problem 22 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{6 y}{y+5}+1\)

Step-by-Step Solution

Answer

The simplified form is \( \frac{7y + 5}{y+5} \).

Step 1: Understand the Expression

We are given the expression \( \frac{6y}{y+5} + 1 \). Our task is to perform the arithmetic operation and simplify it if possible. The expression consists of a rational term \( \frac{6y}{y+5} \) and a whole number 1.

Step 2: Identify a Common Denominator

To simplify \( \frac{6y}{y+5} + 1 \), we need a common denominator. In this case, \( y+5 \) is the denominator of the rational term. To combine the terms, we consider 1 as \( \frac{y+5}{y+5} \), creating an equivalent fraction with a denominator of \( y+5 \).

Step 3: Rewrite the Expression

Rewrite the expression with a common denominator:\[\frac{6y}{y+5} + \frac{y+5}{y+5}.\]

Step 4: Add the Fractions

Since both terms now have the same denominator, we can add them directly:\[\frac{6y + (y+5)}{y+5}.\]

Step 5: Simplify the Numerator

Combine like terms in the numerator:\[6y + y + 5 = 7y + 5.\]

Step 6: Present the Simplified Expression

The simplified expression is:\[\frac{7y + 5}{y+5}.\] Since no further factoring or simplification can be applied, this is our final result.

Key Concepts

Rational Expressions

A rational expression is essentially a fraction in which both the numerator and the denominator are polynomials. These expressions are quite common in algebra and are the building blocks for many other mathematical topics.
The term "rational" comes from the word "ratio." In a rational expression, the ratio of two polynomial expressions is presented. It is crucial to remember that a rational expression is only defined when the denominator is not equal to zero. This means that any values that make the denominator zero are restrictions on the variable.
When working with rational expressions, such as \( \frac{6y}{y+5} \), you need to be vigilant about the values of \( y \) because they cannot make the denominator zero. In this case, \( y+5=0 \) tells us that \( y eq -5 \). Understanding this helps prevent potential division by zero errors.

Arithmetic Operations

Arithmetic operations in the context of rational expressions often involve addition, subtraction, multiplication, or division of fractions. Each operation requires a set of rules to combine and simplify the expressions effectively.
Addition and Subtraction: These operations require a common denominator, just like when working with numerical fractions. In our task, adding \( \frac{6y}{y+5} \) to 1 involved converting 1 into a fraction \( \frac{y+5}{y+5} \) to align the denominators. This step is vital for combining fractions.
Multiplication: You multiply the numerators together and the denominators together to form a new fraction. Always simplify afterwards if possible.
Division: To divide fractions, you multiply by the reciprocal of the divisor. This operation often requires simplification afterward as well.
Mastering these operations ensures you can tackle various rational expressions confidently, simplifying them to make further analysis or application easier.

Simplifying Expressions

Simplifying expressions is all about reducing them to their simplest form. This often involves combining like terms, reducing fractions, and sometimes factoring or expanding expressions.
For our expression \( \frac{6y}{y+5} + 1 \), simplifying involved rewriting the expression with a common denominator and then combining like terms in the numerator. Once the expression \( \frac{6y + y + 5}{y+5} \) was formed, it simplified further to \( \frac{7y + 5}{y+5} \).
To simplify effectively:

• Combine Like Terms: Always look for terms in the expression that can be combined. They often simplify the overall structure.
• Reduce Fractions: If the numerator and denominator have common factors, those can cancel out. Though not used in this task, it is something to consider in other situations.
• Check for Further Simplification: Ensure there are no additional ways to reduce the complexity of the expression once you've performed the initial steps.

Simplification aims to make expressions more manageable and easier to interpret, laying a solid foundation for solving equations or performing additional operations.