find the value of x in: 1/x-a + 1/x-b = 2/x-c

To find the value of \( x \) for the equation 

\[ \frac{1}{x-a} + \frac{1}{x-b} = \frac{2}{x-c} \]

we can solve it step-by-step.

1. **Find a common denominator for the left-hand side of the equation:**

\[ \frac{(x-b) + (x-a)}{(x-a)(x-b)} = \frac{2}{x-c} \]

2. **Simplify the numerator on the left-hand side:**

\[ \frac{(x - b) + (x - a)}{(x-a)(x-b)} = \frac{2}{x-c} \]

\[ \frac{2x - a - b}{(x-a)(x-b)} = \frac{2}{x-c} \]

3. **Cross-multiply to eliminate the fractions:**

\[ (2x - a - b)(x - c) = 2(x - a)(x - b) \]

4. **Expand both sides of the equation:**

\[ (2x - a - b)(x - c) = 2(x^2 - (a+b)x + ab) \]

Expanding the left-hand side:

\[ 2x(x - c) - (a + b)(x - c) = 2x^2 - 2(a + b)x + 2ab \]

\[ 2x^2 - 2xc - ax + ac - bx + bc = 2x^2 - 2(a + b)x + 2ab \]

5. **Simplify and combine like terms:**

Collecting like terms:

\[ 2x^2 - 2xc - ax - bx + ac + bc = 2x^2 - 2ax - 2bx + 2ab \]

Since \( 2x^2 \) cancels out:

\[ -2xc - ax - bx + ac + bc = -2ax - 2bx + 2ab \]

\[ -2xc + ac + bc = -2ax - 2bx + 2ab \]

6. **Group the terms containing \( x \):**

\[ -2xc + ac + bc = -2ax - 2bx + 2ab \]

\[ -2xc = -2ax - 2bx + 2ab - ac - bc \]

7. **Isolate \( x \):**

\[ -2xc + 2ax + 2bx = 2ab - ac - bc \]

\[ x(-2c + 2a + 2b) = 2ab - ac - bc \]

\[ x(2a + 2b - 2c) = 2ab - ac - bc \]

8. **Solve for \( x \):**

\[ x = \frac{2ab - ac - bc}{2a + 2b - 2c} \]

\[ x = \frac{2ab - ac - bc}{2(a + b - c)} \]

Therefore, the value of \( x \) is:

\[ x = \frac{2ab - ac - bc}{2(a + b - c)} \]

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you solve quadratic equations?
2. What are the properties of rational functions?
3. How do you solve equations with logarithms?
4. What methods can be used to solve systems of nonlinear equations?
5. How do you solve exponential equations?

**Tip:** Always check the domain of the original equation to ensure that the solutions do not make any denominators zero.

Learn how to solve the equation 1/(x-a) + 1/(x-b) = 2/(x-c) step-by-step with detailed mathematical explanations. This problem involves rational functions and algebraic manipulation. Suitable for advanced high school students and college-level learners.

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