Problem 34 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{1}{y+5}+\frac{2}{3 y}\)

Step-by-Step Solution

Answer

\(\frac{5(y+2)}{3y(y+5)}\)

Step 1: Identify the Least Common Denominator (LCD)

To add fractions, we need a common denominator. The denominators here are \(y+5\) and \(3y\). The least common denominator is \((y+5) \times 3y\).

Step 2: Rewrite Each Fraction with the LCD

Multiply the numerator and the denominator of each fraction by whatever is needed to give them the LCD as the new denominator.- For \(\frac{1}{y+5}\), multiply by \(3y\), giving \(\frac{3y}{3y(y+5)}\).- For \(\frac{2}{3y}\), multiply by \(y+5\), giving \(\frac{2(y+5)}{3y(y+5)}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add the numerators:\[\frac{3y}{3y(y+5)} + \frac{2(y+5)}{3y(y+5)} = \frac{3y + 2(y+5)}{3y(y+5)}\]

Step 4: Simplify the Numerator

Expand and simplify the numerator:\[3y + 2(y+5) = 3y + 2y + 10 = 5y + 10\]

Step 5: Write the Final Simplified Expression

Combine the simplified numerator with the common denominator:\[\frac{5y + 10}{3y(y+5)}\] Factor the numerator: \(5(y+2)\) making the expression: \[\frac{5(y+2)}{3y(y+5)}\]

Key Concepts

Least Common Denominator

When working with fractions that have different denominators, it is crucial to find a common basis for them, so they can be easily added or subtracted.
This common basis is known as the "Least Common Denominator" (LCD).
The LCD is the smallest expression that both denominators can divide into evenly.

In our exercise, we have the denominators as \(y+5\) and \(3y\).
To find the LCD, you need to multiply these two denominators together.

• The LCD becomes \((y+5) \times 3y\).

Why is it important to find the LCD? Because it allows you to perform operations like adding or subtracting the fractions easily!
By converting the fractions to have the same denominator, you are setting the stage for a straightforward addition operation.
It may seem a bit cumbersome at first, but practice makes it easier.

Adding Fractions

Adding fractions requires the denominators to be the same. Once the LCD is found, you can adjust each fraction so they have this common denominator.
This is done by multiplying both the numerator and the denominator of each fraction by whatever number or expression is necessary to make both denominators equal the LCD.

Let's use the fractions from our problem:

• For \(\frac{1}{y+5}\), multiply by \(3y\) to get \(\frac{3y}{3y(y+5)}\).
• For \(\frac{2}{3y}\), multiply by \(y+5\) to get \(\frac{2(y+5)}{3y(y+5)}\).

Now both fractions have the denominator \(3y(y+5)\).
Once they share the same denominator, you add the numerators together.
Think of it as combining like terms in the numerators.

Simplifying Expressions

After adding the fractions, the next step is to simplify the resulting expression.
This involves simplifying the numerator and looking for common factors between the numerator and the denominator.

In our problem, after adding the numerators, we have:

• Numerator: \(3y + 2(y+5)\)
• After simplifying: \(3y + 2y + 10 = 5y + 10\)

To simplify further, factor out the greatest common factor in the numerator, which is 5 in this case.

• This gives us: \(5(y+2)\)

Finally, present the expression as: \(\frac{5(y+2)}{3y(y+5)}\).
This is your simplified algebraic expression.
Remember, simplifying expressions can help reveal relationships in the equation and make further mathematical operations more manageable.