Solving equations that involve fractions and variables in the denominator can seem challenging at first, but with a few simple steps, you can tackle them effectively.
First, let’s look at an example equation:
Example: Solve the equation:
\( \frac{x}{2} + \frac{3}{x} = 5 \)
1. **Identify the Problem**: Start by observing the fractions and the variable in the denominator. In this case, we have the variables\(x\) in both the numerator and the denominator.
2. **Clear the Denominators**: Multiply every term in the equation by the least common denominator (LCD) to eliminate the fractions. For our problem, the LCD is 2x, since it will eliminate the fractions.
Multiplying the entire equation by 2x gives:
\( 2x \cdot \left(\frac{x}{2}\right) + 2x \cdot \left(\frac{3}{x}\right) = 2x \cdot 5 \)
This results in:
\( x^2 + 6 = 10x \)
3. **Rearrange the Equation**: Next, move all terms to one side to set the equation to zero:
\( x^2 – 10x + 6 = 0 \)
4. **Factor or Use the Quadratic Formula**: Now, try to factor the quadratic equation if possible, or use the quadratic formula. In this case, the quadratic does not factor easily, so we apply 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:
\( x = \frac{10 \pm \sqrt{(-10)^2 – 4 \cdot 1 \cdot 6}}{2 \cdot 1} \)
This simplifies to:
\( x = \frac{10 \pm \sqrt{100 – 24}}{2} = \frac{10 \pm \sqrt{76}}{2} = \frac{10 \pm 2\sqrt{19}}{2} \)
Which becomes:
\( x = 5 \pm \sqrt{19} \)
5. **Check Your Answers**: Finally, substitute the found values back into the original equation to verify they work. Make sure that none of your solutions make any denominators zero.
By following these steps, you can systematically solve equations involving fractions and variables in the denominator. Practice with different equations to build your confidence.