Problem 107 - College Algebra with Modeling and Visualization

Question

Clear fractions and solve. $$ \frac{2 x}{9-x^{2}}+\frac{1}{3-x}=0 $$

Step-by-Step Solution

Answer

The solution is \( x = -1 \).

Step 1: Identify the Fractions

We start by identifying the fractions in the equation: \( \frac{2x}{9-x^2} \) and \( \frac{1}{3-x} \). Our goal is to clear these fractions by finding a common denominator.

Step 2: Find the Common Denominator

The denominators are \( 9 - x^2 \) and \( 3 - x \). Notice that \( 9 - x^2 = (3-x)(3+x) \). Therefore, the common denominator is \((3-x)(3+x)\).

Step 3: Multiply Through by the Common Denominator

Multiply each term by \((3-x)(3+x)\):\[(3-x)(3+x) \cdot \frac{2x}{9-x^2} + (3-x)(3+x) \cdot \frac{1}{3-x} = 0\] This simplifies to:\[2x + (3+x) = 0\] After cancellation.

Step 4: Simplify the Equation

Simplify the equation:\[2x + 3 + x = 0\] Combine like terms to get:\[3x + 3 = 0\]

Step 5: Solve for x

Remove 3 from both sides:\[3x = -3\]Then, divide both sides by 3:\[x = -1\]

Key Concepts

Understanding Fractions Through an Example

In our exercise, we are dealing with the equation \( \frac{2x}{9-x^2} + \frac{1}{3-x} = 0 \). Here, fractions are mathematical expressions that represent the division of two quantities. In this problem, the fractions \( \frac{2x}{9-x^2} \) and \( \frac{1}{3-x} \) need to be addressed to make the equation easier to solve. The top part of a fraction is called the numerator, and the bottom part is the denominator.

Recognizing and working with fractions is crucial in many areas of college algebra. When dealing with equations that include fractions, our primary goal is often to "clear" them by eliminating or simplifying them. This is facilitated by finding a common denominator, which allows us to transform the equation into a simpler form.

By clearing fractions, not only do we make the arithmetic simpler, but we also bring the focus back to solving the equation effectively.

Finding a Common Denominator

In order to eliminate fractions in an equation, finding a common denominator is a key step. For our specific problem, the denominators are \( 9-x^2 \) and \( 3-x \). A common denominator is essentially a shared multiple, which in this case, helps in combining the fractions.

Let’s break it down:

• The expression \( 9-x^2 \) can be factored as \((3-x)(3+x)\).
• Thus, the common denominator for our fractions is \((3-x)(3+x)\).

Using a common denominator for all fractions in an equation allows us to focus on simpler expressions without fractions. Essentially, by multiplying through by this common denominator, we effectively "clear" the fractions.

This approach not only simplifies the equation but also prepares it for traditional algebraic solving techniques, such as combining like terms.

Solving Equations After Clearing Fractions

Once the fractions are cleared by using the common denominator, the process of solving the equation becomes straightforward. For the exercise equation, multiplying by the common denominator \((3-x)(3+x)\) reduces the fractions and simplifies the expression to \(2x + (3+x) = 0\).

The next step is to simplify by combining like terms:

• This gives us \(2x + 3 + x = 0\), which simplifies to \(3x + 3 = 0\).
• Subtract 3 from both sides to isolate terms containing \(x\): \(3x = -3\).
• Finally, divide both sides by 3 to solve for \(x\): \(x = -1\).

Breaking this process into small, manageable steps helps in achieving clarity when solving equations involving fractions. Furthermore, understanding each step makes it easier to apply these methods to other algebraic equations.