Problem 55 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ \frac{2 y+1}{y+4}-\frac{y+6}{y+1} $$

Step-by-Step Solution

Answer

Question: Subtract the following algebraic fractions: $$\frac{2y+1}{y+4} - \frac{y+6}{y+1}$$ Answer: $$\frac{y^2-7y-23}{(y+4)(y+1)}$$

Step 1: Find the least common denominator (LCD)

To find the least common denominator, we need to find the lowest common multiple (LCM) of the denominators \((y+4)\) and \((y+1)\). Since they are both linear expressions and have no common factors, their LCM is just their product. Therefore, the LCD is \((y+4)(y+1)\).

Step 2: Rewrite the fractions with the same denominator

To rewrite the given fractions with \((y+4)(y+1)\) as their denominator, multiply the numerator and denominator of the first fraction by \((y+1)\) and the numerator and denominator of the second fraction by \((y+4)\). $$ \frac{2y+1}{y+4}\cdot\frac{y+1}{y+1} - \frac{y+6}{y+1}\cdot\frac{y+4}{y+4} $$ Now, expand the numerators: $$ \frac{(2y+1)(y+1)}{(y+4)(y+1)} - \frac{(y+6)(y+4)}{(y+1)(y+4)} $$

Step 3: Perform the subtraction

Since the two fractions have the same denominator, we can perform the subtraction by simply subtracting their numerators. $$ \frac{(2y+1)(y+1) - (y+6)(y+4)}{(y+4)(y+1)} $$ Now, expand the numerators and simplify them: $$ \frac{2y^2+3y+1 - (y^2 +10y + 24)}{(y+4)(y+1)} $$ Subtract the terms within the parenthesis and combine like terms: $$ \frac{2y^2+3y+1- y^2 -10y - 24}{(y+4)(y+1)} $$ $$ \frac{y^2-7y-23}{(y+4)(y+1)} $$ The simplified expression for the given problem is $$ \frac{y^2-7y-23}{(y+4)(y+1)} $$

Key Concepts

Least Common Denominator

When working with algebraic fractions, finding the least common denominator (LCD) is a crucial step in performing operations such as addition and subtraction. The LCD is the smallest expression that each of the denominators in a set of fractions can divide into without leaving a remainder.

To identify the least common denominator in algebraic terms, we look for the least common multiple (LCM) of their denominators. If the denominators are linear expressions that do not share any common factors—like in the provided exercise, where the denominators are \(y+4\) and \(y+1\)—the LCD is simply their product. So, we have \(LCD = (y+4)(y+1)\).

Knowing the LCD is essential for combining fractions, as it allows us to rewrite each fraction with a common denominator, facilitating the subtraction or addition of the numerators. Be watchful that each numerator is accurately multiplied by the necessary factor to align with the new denominator.

Linear Expressions

A linear expression is an algebraic statement that has variables raised to the first power, constants, and coefficients. They usually take the form \(ax + b\), where \(a\) and \(b\) are constants, and \(x\) is a variable. In our exercise, \(y+4\) and \(y+1\) are both examples of linear expressions.

Characteristics of Linear Expressions

• No exponents higher than 1
• Can have one or more terms
• Often found as parts of polynomial equations or inequalities

Understanding how to manipulate linear expressions is fundamental in algebra. This includes being able to expand, factor, and simplify them, which enables us to solve equations and simplify algebraic fractions.

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra that involves reducing an expression to its simplest form. It most often includes expanding products, combining like terms, and canceling common factors in numerators and denominators.

With algebraic fractions, once a common denominator is established, we can focus on simplifying the expression by combining the numerators. This usually requires distributing products in the numerators and then subtracting or adding terms accordingly. Simplification serves to minimize the complexity of algebraic expressions, thus making it easier to understand them and further manipulate or solve equations where they appear. In our exercise, after subtracting and combining like terms, we simplified the expression to \(\frac{y^2-7y-23}{(y+4)(y+1)}\).

Polynomial Subtraction

Polynomial subtraction is similar to polynomial addition but involves the additional step of changing the sign of each term of the polynomial that we are subtracting. After ensuring that algebraic fractions have a common denominator, the process requires the subtraction of their numerators, term by term.

Following the steps used in the exercise:

1. Align the like terms—which have the same variable raised to the same power.
2. Change the signs of the terms of the polynomial being subtracted.
3. Combine like terms resulting in a simpler polynomial or expression.

The goal is to condense the expression into its most reduced form, free of any parentheses and as few terms as possible, just like we achieved in the expression \(\frac{y^2-7y-23}{(y+4)(y+1)}\).