Problem 3 - Algebra 1: Concepts and Skills

Question

Add. Simplify your answer. $$ \frac{8 y}{y+3}+\frac{10-3 y}{y+3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5(y + 2)}{y+3} \)

Step 1: Identify Common Denominator

The two fractions have the same denominator \((y+3)\), this means they can be added directly.

Step 2: Add the Numerators

Add the numerators of the two fractions as follows: \[ \frac{8 y + (10 - 3 y)}{y+3} \].

Step 3: Simplify the Numerator

Simplify the numerators by combining the 'y' terms: \[ \frac{8 y - 3 y + 10}{y+3} = \frac{5 y + 10}{y+3} \].

Step 4: Factorize Numerator

Factorize the numerator to find any common factors with the denominator. Here, 5 is a common factor: \[ \frac{5(y + 2)}{y+3} \]

Step 5: Check for further Simplification

The fraction cannot be further simplified as there are no common factors between the numerator and denominator.

Key Concepts

Common Denominator

When adding or subtracting fractions, having a common denominator is crucial. The common denominator means the bottom part of the fractions is the same, allowing you to combine the fractions easily. In the given exercise, both fractions, \( \frac{8y}{y+3} \) and \( \frac{10-3y}{y+3} \), already have the same denominator, \( y+3 \). This simplifies the process since you can directly add the numerators without modifying the denominators. To ensure effective addition:

• Verify if both fractions have the same denominator.
• If they do, proceed to add or subtract the numerators.
• If not, find a common denominator by identifying a common multiple.

By checking the denominators first, you can make the adding process smoother.

Simplifying Fractions

Simplifying fractions means rewriting them in their simplest form. After combining the numerators, you often need to simplify, as shown in the exercise solution: 1. First, simplify the expression by combining like terms. In this exercise: \(8y - 3y + 10 = 5y + 10\).2. A critical aspect of simplification is to check for any common factors. In the numerator \(5y + 10\), 5 is a common factor.After factorization, it appears as \(5(y + 2)\). At this stage, the simplified fraction is \(\frac{5(y+2)}{y+3}\). Ensure to always:

• Reduce common factors from both the numerator and the denominator.
• Write in the simplest terms whenever possible.

This ensures clarity and simplicity in expressing the final results.

Factorization

Understanding factorization is essential in algebra, especially when simplifying expressions. Factorization involves breaking down an algebraic expression into simpler 'factors' that, when multiplied, give the original expression. In our exercise:- After simplifying \(5y + 10\), observe that both terms share a common factor: 5.- Hence, \(5y + 10\) can be factorized into \(5(y + 2)\).Factorization becomes useful when you check a fraction for further simplification. Here are some tips:

• Identify common factors in the terms of the numerator or the denominator.
• Extract the common factors for better clarity and potential simplification.
• Always consider the possibility of factoring quadratic terms or expressions within polynomials.

Mastering factorization helps you in transforming and dealing with complex algebraic expressions efficiently.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. They are a fundamental part of algebra and play a significant role in various mathematical procedures. In the discussed exercise:- The expressions \(8y\) and \(10 - 3y\) are part of an algebraic expression that involves addition.- Understanding algebraic expressions involves recognizing terms and operations such as addition, subtraction, and multiplication within expressions.Here's how to effectively manage algebraic expressions:

• Identify different parts of an expression, such as coefficients, variable terms, and constants.
• Pay attention to operations that connect different terms, as they define the expression's structure.
• Practice combining like terms to simplify expressions more efficiently, which is crucial in finding solutions.

Developing strong skills with algebraic expressions helps you solve equations and manipulate algebraic formulas with greater ease and accuracy.