Problem 47 - Elementary Algebra

Question

For the following problems, replace \(N\) with the proper quantity. $$ \frac{3 b}{b-3}=\frac{N}{b^{2}-11 b+24} $$

Step-by-Step Solution

Answer

Answer: The proper quantity to replace N with is \(3b^2 - 24b\).

Step 1: Factor the Quadratic Expression in the Denominator

First, we'll factor the quadratic expression in the denominator which is \(b^2 - 11b + 24\). This can be factored as \((b - 3)(b - 8)\), since \(-3 \cdot -8 = 24\) and \(-3 + -8 = -11\). So now our equation looks like this: $$ \frac{3b}{b-3} = \frac{N}{(b-3)(b-8)} $$

Step 2: Simplify the Fractional Equalities

Now that we have factored the denominator, let's see if there's a common factor between the numerator and the denominator that we can cancel to further simplify the equation. As we can see, there is a common factor \((b-3)\) in both the numerators and denominators, so we can simplify the given equation as follows: $$ \frac{3b}{b-3} \times \frac{1}{b-8} = \frac{N}{(b-3)(b-8)} \times \frac{1}{b-8} $$ Removing the common factor, we get: $$ \frac{3b}{1} = \frac{N}{b-8} $$

Step 3: Determine the Proper Quantity for N

Now, we can easily find the proper quantity for N. Multiplying both sides of the equation by \((b-8)\), we have: $$ N = 3b(b-8) $$ Now, we can simplify the expression for N: $$ N = 3b^2 - 24b $$ So, the proper quantity to replace N with is \(3b^2 - 24b\). Our final simplified equation is: $$ \frac{3b}{b-3} = \frac{3b^2 - 24b}{b^2 - 11b + 24} $$

Key Concepts

simplifying algebraic fractions

Simplifying algebraic fractions involves breaking down a fraction to its simplest form. It means reducing the fraction by removing common factors from the numerator and the denominator.
This process requires us to factor expressions, especially when variables are involved. Let’s consider an algebraic expression like \( \frac{3b}{b-3} \). To simplify such a fraction:

• First, look for common factors between the numerator and the denominator.
• If both the numerator and the denominator share a common factor, cancel it out.

For example, if you have \( \frac{3b}{b-3} \) and \( \frac{N}{(b-3)(b-8)} \), observe the common term \((b-3)\) in the denominator. This allows simplification.
By removing the common factor, you transform the equation into a simpler form. This makes it easier to solve or substitute values. Simplifying helps make complex equations more manageable.

algebraic expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operational symbols. They represent a range of possible values and are used extensively in equations and formulae.
For instance, in the expression \(3b^2 - 24b\), the components include:

• **Numbers:** These are constants, like 3 and 24, which are coefficients in this case.
• **Variables:** These represent unknown values, such as \(b\).
• **Operations:** Includes multiplication and subtraction signs, showing mathematical operations.

Understanding algebraic expressions involves knowing the terms, constants, and variables. Recognizing how to manipulate them, such as factoring or combining like terms, is often key?
With algebraic expressions, you can solve quadratic equations, factor polynomials, and analyze rational equations. They form the basis of algebra, enabling us to express relationships and solve problems systematically.

rational equations

Rational equations are equations that involve fractions with polynomials in the numerator and the denominator. These types of equations are prevalent in algebra because they enable complex relationship modeling between variables.
For example, the equation \(\frac{3b}{b-3} = \frac{N}{b^2 - 11b + 24} \) is a rational equation.
To solve rational equations, follow these steps:

• **Identify common denominators:** Find a common multiple of the denominators.
• **Clear the fractions:** Multiply through by the common denominator to eliminate the fractions.
• **Solve the resulting equation:** With the fractions gone, solve the equation using algebraic methods.

It's essential to check your solutions by substituting them back into the original equation. This helps ensure they don’t result in undefined expressions, such as division by zero.
Rational equations often turn up in real-world scenarios, such as calculating rates and ratios, making it important to understand the principles behind them.