How can I effectively solve equations that involve fractions and variables in the denominator?

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Solving equations with fractions and variables in the denominator can be a bit tricky, but with a systematic approach, you can tackle these problems confidently. Here’s a detailed step-by-step method to help you:

Step 1: Identify the Equation

Start by clearly identifying the equation you need to solve. For instance, you might have something like:

\( \frac{x}{2} + \frac{3}{x} = 5 \)

Step 2: Find the Least Common Denominator (LCD)

Next, you’ll want to determine the Least Common Denominator (LCD) of all the fractions in the equation. In our example, the denominators are 2 and x. The LCD is 2x.

Step 3: Multiply Through by the LCD

Now, multiply every term in the equation by the LCD to eliminate the fractions. This gives us:

2x \cdot \left( \frac{x}{2} \right) + 2x \cdot \left( \frac{3}{x} \right) = 2x \cdot 5

When simplified, we get:

x^2 + 6 = 10x

Step 4: Rearrange the Equation

Now, rearrange the equation by moving all terms to one side:

x^2 - 10x + 6 = 0

Step 5: Solve the Quadratic Equation

At this point, you can either factor the quadratic equation, complete the square, or use the quadratic formula. In this case, let’s use the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Here, a = 1, b = -10, and c = 6. Plugging in these values gives us:

x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}

Calculating this:

x = \frac{10 \pm \sqrt{100 - 24}}{2} = \frac{10 \pm \sqrt{76}}{2} = \frac{10 \pm 2\sqrt{19}}{2} = 5 \pm \sqrt{19}

Step 6: Check Your Solutions

Finally, always check your solutions by substituting them back into the original equation to ensure they work, especially since we multiplied through by the LCD, which can potentially introduce extraneous solutions.

In summary, solving equations with fractions and variables in the denominator involves:

  • Identifying the equation
  • Finding the LCD
  • Multiplying through by the LCD
  • Rearranging to form a quadratic equation
  • Using the quadratic formula to find solutions
  • Checking your solutions for validity

With practice, you’ll find that solving these types of equations becomes easier and more intuitive!

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