Problem 9 - Elementary and Intermediate Algebra

Question

By what should both sides of the equation be multiplied to clear it of fractions? $$ \text { a. } \frac{1}{y}=20-\frac{5}{y} $$ $$ \text { b. } \frac{x}{x^{2}-4}=\frac{4}{x-2} $$

Step-by-Step Solution

Answer

a) Multiply by \(y\); b) Multiply by \((x-2)(x+2)\).

Step 1: Understanding the Problem

For part a, eliminate the fractions by determining the least common multiple (LCM) of the denominators: \(y\). For part b, calculate the LCM of the denominators: \(x^2-4\) and \(x-2\). These denominators can be factored to help find the LCM.

Step 1 for Part a: Identify the Denominator

The denominators in the equation \( \frac{1}{y}=20-\frac{5}{y} \) are both \(y\).

Step 2 for Part a: LCM of Denominators

The least common multiple of the denominators \(y\) and \(y\) is \(y\) itself.

Step 3 for Part a: Multiply Both Sides

To clear the equation of fractions, multiply every term by \(y\). This results in the equation becoming: \(1 = 20y - 5\).

Step 1 for Part b: Factor the Denominators

The denominators are \(x^2-4\) and \(x-2\). Factoring gives \((x-2)(x+2)\) and \(x-2\), respectively.

Step 2 for Part b: LCM of Denominators

The LCM of \((x-2)(x+2)\) and \(x-2\) is \((x-2)(x+2)\), since it includes the term \(x-2\) needed for both fractions.

Step 3 for Part b: Multiply Both Sides

Multiply every term by \((x-2)(x+2)\) to eliminate the fractions. The resulting equation is: \(x = 4(x+2)\).

Key Concepts

Fractions in Equations

Fractions in equations can make solving them seem a bit tricky, but remember, their presence is nothing to worry about. The key is to eliminate them to simplify the equation. In equations with fractions, each term of the equation typically has a denominator. Our goal is to find a common denominator so we can get rid of these fractions by multiplying each term by it.

• Identify all the denominators in your equation.
• Determine the least common multiple (LCM) of these denominators.
• Multiply every term in the equation by this LCM to clear the fractions away entirely.

By following these steps, you transform an equation laden with fractions into a much simpler form, which is typically easier to solve. As demonstrated in the original exercise, we eliminated fractions from the given equations, making them significantly easier to handle.

Least Common Multiple

When working with equations that involve denominators, finding the least common multiple (LCM) of these denominators is essential. The LCM is the smallest multiple that is shared by each of the denominators. Its purpose is to enable you to eliminate all fractions in one go.

To determine the LCM:

• Factorize each of the denominators fully. This might involve breaking down polynomials if necessary.
• Identify the largest power of every factor present in any of the denominators.
• Multiply these factors together. The product is your LCM.

For instance, in part b of the exercise, we found the LCM as \( (x-2)(x+2) \) because it contains all the factors needed to represent both denominators. Once the LCM is determined, multiplying through by this value clears the denominators effectively.

Factoring Polynomials

Factoring polynomials is a handy skill when dealing with algebraic equations, especially those involving fractions. This process involves rewriting a polynomial as a product of its factors, which can reveal common terms.

Let's take part b from the exercise as an example. Here, the polynomial denominators were \( x^2-4 \) and \( x-2 \). Notice that \( x^2-4 \) can be factored into \( (x-2)(x+2) \), a process known as factoring a difference of squares. Once you have these factors:

• You'll see that \( x-2 \) is a common factor, allowing you to easily determine the LCM.
• Reduce complexity in the equation by canceling identical terms when possible.

Understanding how to factor polynomials not only aids in finding the LCM but also simplifies solving equations, making it a valuable tool in your algebra toolkit.