Problem 47 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2y^2 + y}{(2y+3)^2} \).

Step 1: Identify a Common Denominator

The terms in the expression \( \frac{y-1}{2y+3} + \frac{3}{(2y+3)^2} \) have different denominators. The first term has a denominator of \( 2y+3 \), and the second term has a denominator of \( (2y+3)^2 \). Therefore, the least common denominator (LCD) is \( (2y+3)^2 \).

Step 2: Rewrite Each Fraction with the Common Denominator

To write both fractions with the LCD \((2y+3)^2\), multiply the numerator and denominator of the first fraction by \(2y+3\):\[\frac{y-1}{2y+3} \times \frac{2y+3}{2y+3} = \frac{(y-1)(2y+3)}{(2y+3)^2}\]The second fraction already has the denominator \((2y+3)^2\):\[\frac{3}{(2y+3)^2} = \frac{3}{(2y+3)^2}\]

Step 3: Combine the Fractions

Now that both fractions have the same denominator, they can be added together:\[\frac{(y-1)(2y+3)}{(2y+3)^2} + \frac{3}{(2y+3)^2} = \frac{(y-1)(2y+3) + 3}{(2y+3)^2}\]

Step 4: Simplify the Numerator

Expand and simplify the expression in the numerator:\[(y-1)(2y+3) = y(2y+3) - 1(2y+3) = 2y^2 + 3y - 2y - 3 = 2y^2 + y - 3\]Now, write the complete numerator:\[2y^2 + y - 3 + 3 = 2y^2 + y\]

Step 5: Write Final Answer

The combined and simplified fraction is:\[\frac{2y^2 + y}{(2y+3)^2}\]

Key Concepts

Fractions

Fractions are a way to represent parts of a whole, much like slicing a cake into pieces. In algebra, fractions come with variables. For example, in the fraction \(\frac{y-1}{2y+3}\), the numerator (\(y-1\)) is the part above the line, and the denominator (\(2y+3\)) is the part below.
When dealing with fractions, our goal is often to make them simpler or to combine them. This is especially true in algebra, where finding a common denominator helps us tackle both operations at once.
Algebraic fractions can be tricky due to variables, but the same arithmetic rules for fractions apply.

• You add or subtract fractions by making the denominators the same.
• Multiplication involves multiplying numerators together and denominators together.
• For division, you "flip" (take the reciprocal of) the second fraction and multiply.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions. In our exercise, the denominators were \(2y+3\) and \((2y+3)^2\).
The least common denominator (LCD) serves as a shared base for both fractions, allowing them to be combined.
Here's how to find a common denominator:

• Identify the unique factors in each denominator.
• Determine the highest power of these factors.
• Construct the LCD from these factors.

For example, in our case, \((2y+3)\) is multiplied with itself to match the second fraction's \((2y+3)^2\). This adjustment ensures both fractions are working from the same base.

Algebraic Simplification

Algebraic simplification reduces complex expressions to simpler forms. After finding a common denominator, the next step is to simplify expressions in numerators.
In the exercise, once the fractions are combined, the expression \((y-1)(2y+3)\) gets expanded to distribute the terms.
Breaking it down step by step:

• First multiply \(y\) by each term in \(2y+3\).
• Next multiply \(-1\) by each term in \(2y+3\).
• Combine like terms to arrive at a simpler expression.

Remember, simplifying might reveal opportunities to further cancel terms between numerators and denominators, but only if factors are common.

Rational Equations

Rational equations involve fractions whose numerators and denominators contain algebraic expressions. The goal is often to solve for a variable or simplify the expression.
In our exercise, we simplified a rational expression, striking a balance between making operations clearer and keeping everything in the simplest form possible.
Key strategies include:

• Avoiding division by zero by considering denominators.
• Using common denominators for smooth addition or subtraction.
• Consistently simplifying numerators and denominators throughout the process.

Approaching rational equations requires patience and a firm grasp of algebraic manipulation, as is practiced in this type of problem.