Problem 29 - Algebra A Combined Function

Question

Solve each equation. $$ \frac{11}{2 x}+\frac{2}{3}=\frac{7}{2 x} $$

Step-by-Step Solution

Answer

The solution is \( x = -3 \).

Step 1: Understanding the Equation

The equation given is \( \frac{11}{2x} + \frac{2}{3} = \frac{7}{2x} \). We see that the terms have fractions with \(2x\), so we want to simplify the equation by removing these fractions if possible.

Step 2: Isolate Like Terms

Subtract the term \( \frac{7}{2x} \) from both sides of the equation to group the terms that contain \(x\): \( \frac{11}{2x} - \frac{7}{2x} + \frac{2}{3} = 0 \).

Step 3: Simplify the Fraction Group

Combine the terms with \( \frac{11}{2x} - \frac{7}{2x} = \frac{4}{2x} \). This simplifies to \( \frac{2}{x} \). Now the equation is \( \frac{2}{x} + \frac{2}{3} = 0 \).

Step 4: Eliminate the Fraction

To eliminate the fraction of \( \frac{2}{x} \), multiply each term by \(x\), resulting in \( 2 + \frac{2x}{3} = 0 \).

Step 5: Solve for x

Subtract 2 from both sides to isolate the term with \(x\): \( \frac{2x}{3} = -2 \). Then multiply both sides by 3: \( 2x = -6 \).

Step 6: Find the Value of x

Divide both sides by 2 to solve for \(x\): \( x = -3 \).

Key Concepts

Fractional Equations

Fractional equations are equations that contain fractions with variables in the denominator. These types of equations may seem a bit tricky at first, but with practice, they can be easily tackled. They often require special attention to the variables, as they can appear in the denominator, affecting the entirety of the equation differently than they would in standard linear equations. Here's how you can manage them effectively:

• Identify the portions of the equation that have variables in the denominator.
• Consider an approach to eliminate the fractions, such as finding a common denominator or multiplying throughout by the variable’s denominator.

This helps in creating a simpler equation, which is easier to handle as you move towards finding the solution.

Isolation of Terms

Isolation is a critical process in solving equations, ensuring you're focusing on the variable of interest. The goal is to isolate terms that contain the variable on one side of the equation and constants on the other. To isolate, you can:

• Apply operations like addition, subtraction, multiplication, or division.
• Ensure you're systematically moving terms across the equation to shift the variable terms together.

In the given problem, isolating involved arranging terms with the variable in the denominator into a single group, which simplified further manipulation.

Simplification

Simplification is a powerful tool that helps make complex equations more manageable. By focusing on combining like terms and performing basic arithmetic operations, you can reduce the equation’s complexity. During simplification:

• Combine similar terms, such as those with the same denominator.
• Look for common factors in fractions that can be simplified.

A simplified equation is easier to solve, as it often reveals the underlying structure of the problem, providing clearer steps towards a solution.

Variable Elimination

Variable elimination involves removing the variables from fractions or scenarios where they complicate the solution process. To aid elimination:

• Multiply throughout by the variable’s denominator to rid the equation of fractions.
• Ensure all operations maintain the equation’s balance, taking careful steps if additional variables appear as cancellations might occur that simplify further.

In our example, multiplying the entire equation by the variable allowed removal of fractional components, thereby transforming it into a linear equation that is much simpler to solve.