Problem 41
Question
For the following problems, replace \(N\) with the proper quantity. $$ \frac{a+1}{a-3}=\frac{N}{(a-3)(a-4)} $$
Step-by-Step Solution
Verified Answer
Answer: The proper quantity 'N' to replace in the equation is \(a^2 - 3a - 4\).
1Step 1: Identify the common denominator
We need to find the equivalent fraction with the denominator \((a-3)(a-4)\). Since the given denominator is \((a-3)\), we need to find a relationship between these two denominators.
2Step 2: Multiply the given fraction by the missing factor
We can see that \((a-3)(a-4)\) is the product of \((a-3)\) and the missing factor \((a-4)\). So, to make the denominators equal we need to multiply both the numerator and the denominator of the given fraction by \((a-4)\).
3Step 3: Multiply the given fraction
Now, rewrite the given fraction with the new denominator and multiply the numerator by \((a-4)\) :
$$
\frac{(a+1)(a-4)}{(a-3)(a-4)} = \frac{a^2 - 4a + a - 4}{(a-3)(a-4)}
$$
Simplify the numerator:
$$
\frac{a^2 - 3a - 4}{(a-3)(a-4)}
$$
4Step 4: Identify the value of 'N'
Comparing the simplified fraction with the original equation, we have:
$$
\frac{a^2 - 3a - 4}{(a-3)(a-4)} = \frac{N}{(a-3)(a-4)}
$$
From the equation, we can identify the value of 'N':
$$
N = a^2 - 3a - 4
$$
So, the proper quantity of 'N' to replace in the equation is \(a^2 - 3a - 4\).
Key Concepts
Common DenominatorAlgebraic FractionsSimplifying Expressions
Common Denominator
Understanding the concept of a common denominator is crucial when it comes to working with fractions. This is especially important when the task is to combine fractions that have different denominators or to compare their values.
In algebra, finding the common denominator involves identifying the least common multiple (LCM) between two or more denominators, which allows you to convert the fractions into equivalent ones with the same denominator. To apply this concept, consider when you have two algebraic fractions, such as \(\frac{3}{x-2}\) and \(\frac{4}{x+5}\). The common denominator for these fractions would be the product \( (x-2)(x+5) \), representing the LCM of the original denominators.
To rewrite fractions with the common denominator, multiply the numerator and the denominator of the original fractions by the factors they are missing. This does not change their value—it merely expresses them in a form that is easier to work with during addition, subtraction, or comparison of fractions.
In algebra, finding the common denominator involves identifying the least common multiple (LCM) between two or more denominators, which allows you to convert the fractions into equivalent ones with the same denominator. To apply this concept, consider when you have two algebraic fractions, such as \(\frac{3}{x-2}\) and \(\frac{4}{x+5}\). The common denominator for these fractions would be the product \( (x-2)(x+5) \), representing the LCM of the original denominators.
To rewrite fractions with the common denominator, multiply the numerator and the denominator of the original fractions by the factors they are missing. This does not change their value—it merely expresses them in a form that is easier to work with during addition, subtraction, or comparison of fractions.
Algebraic Fractions
When dealing with algebraic fractions, it's important to recognize they follow the same rules as numerical fractions but instead contain algebraic expressions in the numerator, the denominator, or both. Simplification of these expressions often involves factoring and reducing, similar to how you would simplify a numerical fraction.
For example, the algebraic fraction \(\frac{x^2+x-6}{x^2-9}\) can be simplified by factoring both the numerator and denominator into \(\frac{(x+3)(x-2)}{(x+3)(x-3)}\) and then canceling out the common factor \(x+3\) to get \(\frac{x-2}{x-3}\).
Algebraic fractions can often represent complex relationships, and identifying common factors or terms within them can be key to simplification. In more advanced mathematics, algebraic fractions are essential components of calculus and other higher-level concepts.
For example, the algebraic fraction \(\frac{x^2+x-6}{x^2-9}\) can be simplified by factoring both the numerator and denominator into \(\frac{(x+3)(x-2)}{(x+3)(x-3)}\) and then canceling out the common factor \(x+3\) to get \(\frac{x-2}{x-3}\).
Algebraic fractions can often represent complex relationships, and identifying common factors or terms within them can be key to simplification. In more advanced mathematics, algebraic fractions are essential components of calculus and other higher-level concepts.
Simplifying Expressions
The process of simplifying expressions is a cornerstone of algebra and precalculus. It involves altering algebraic expressions into their simplest form without changing their value. This could mean combining like terms, factoring, expanding expressions, or canceling common factors in fractions.
A simple example is combining like terms in an expression such as \(2x + 3x - x\), which simplifies to \(4x\). In working with algebraic fractions, simplifying might involve expanding the numerator and the denominator and then reducing the fraction to its simplest form as in the exercise example:
\[ \frac{a^2 - 4a + a - 4}{(a-3)(a-4)} \]\
After combining like terms in the numerator, we're left with a simpler expression that accurately represents the original equation, without extraneous components.
Simplifying an expression paves the way for solving equations and understanding the relationship between the variables within an equation. Mastering this skill is essential for success in algebra and beyond.
A simple example is combining like terms in an expression such as \(2x + 3x - x\), which simplifies to \(4x\). In working with algebraic fractions, simplifying might involve expanding the numerator and the denominator and then reducing the fraction to its simplest form as in the exercise example:
\[ \frac{a^2 - 4a + a - 4}{(a-3)(a-4)} \]\
After combining like terms in the numerator, we're left with a simpler expression that accurately represents the original equation, without extraneous components.
Simplifying an expression paves the way for solving equations and understanding the relationship between the variables within an equation. Mastering this skill is essential for success in algebra and beyond.
Other exercises in this chapter
Problem 41
For the following problems, solve the rational equations. $$ \frac{2}{a-5}-\frac{4 a-2}{a^{2}-6 a+5}=\frac{-3}{a-1} $$
View solution Problem 41
For the following problems, perform the multiplications and divisions. $$ 2 a^{5} \div \frac{6 a^{2}}{4 b} $$
View solution Problem 41
For the following problems, add or subtract the rational expressions. $$ \frac{3 y+4}{y+8}-\frac{2 y-5}{y+8} $$
View solution Problem 41
For the following problems, reduce each rational expression to lowest terms. $$ \frac{x^{2}+3 x-10}{x^{2}+2 x-15} $$
View solution