Problem 33 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{y+5}+\frac{3}{4 y}$$

Step-by-Step Solution

Answer

The result of the subtraction of the two given fractions is \(- \frac{1}{y}\).

Step 1: Identify Similar Denominators

Our two fractions, \(\frac{3 y^{2}-1}{3 y^{3}}\) and \(\frac{6 y^{2}-1}{3 y^{3}}\), have similar denominators, so we can directly subtract the numerators.

Step 2: Subtract Numerators

Subtract the numerators of the two fractions: \( (3 y^{2} - 1) - (6 y^{2} - 1)\)

Step 3: Simplify the Expression

Simplify the result to get \(-3 y^{2}\)

Step 4: Write Fraction with Simplified Numerator

The fraction with the new numerator and the same denominator is \(\frac{-3 y^{2}}{3 y^{3}}\)

Step 5: Simplify Fraction

Simplify the fraction by dividing the numerator and the denominator by \(3y^2\) to get \(-\frac{1}{y}\)

Key Concepts

Adding and Subtracting Fractions

When working with fractions, knowing how to add or subtract them is essential, especially when they have the same denominator. Just like in the exercise, when fractions have the same denominator, such as \( \frac{3y^2 - 1}{3y^3} \) and \( \frac{6y^2 - 1}{3y^3} \), you can streamline the process:

• First, keep the common denominator and focus on the numerators.
• Subtract the numerators: \((3y^2 - 1) - (6y^2 - 1)\).
• Be mindful of negative signs—subtracting a positive becomes adding a negative.

Once you've subtracted, you get a new expression for the numerator, which in this case simplifies to \(-3y^2\). Finally, plug this back over the common denominator.

Simplifying Rational Expressions

Simplifying rational expressions involves breaking down the fractions to their simplest form. After performing the subtraction in the previous step, you often have both numerators and denominators that can be simplified:

• Write out the fraction with the new numerator: \(\frac{-3y^2}{3y^3}\).
• Factor out common terms in both the numerator and the denominator. Here, both can be divided by \(3y^2\).
• Perform the division: \(-3y^2 \div 3y^2 = -1\), and \(3y^3 \div 3y^2 = y\).

You end up with \(-\frac{1}{y}\). A well-simplified expression makes it easier to understand and use in further calculations.

Polynomials

Polynomials are expressions consisting of variables and coefficients, involving operations of addition, subtraction, and multiplication.
For example, \(3y^2 - 1\) and \(6y^2 - 1\) are both polynomials. When dealing with these, it's important to:

• Identify like terms to simplify expressions.
• Be cautious with operations, especially to avoid mistakes with subtraction.
• Factor polynomials when possible to ease both simplification and solving.

Understanding polynomials is crucial, as they form the basis of many algebraic structures and processes. Each term in a polynomial is separated by either a plus (+) or a minus (-) sign, making them both versatile and powerful in algebra.