find the value of x: 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 proceed as follows:

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:

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

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

5. Simplify and combine like terms:

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

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

6. Equate the coefficients of \( x \) and the constant terms:

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

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

7. For the \( x \)-term coefficients to be equal, we get:

\[ a + b + 2c = 2(a + b) \]

\[ a + b + 2c = 2a + 2b \]

\[ 2c = a + b \]

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

8. Substitute \( c = \frac{a + b}{2} \) into the equation for the constants:

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

\[ (a + b)\left(\frac{a + b}{2}\right) = 2ab \]

\[ \frac{(a + b)^2}{2} = 2ab \]

\[ (a + b)^2 = 4ab \]

\[ a^2 + 2ab + b^2 = 4ab \]

\[ a^2 - 2ab + b^2 = 0 \]

\[ (a - b)^2 = 0 \]

\[ a = b \]

So, if \( a = b \), then the original equation holds true for any \( x \neq a, b \). Therefore, \( x \) can be any value except \( x \neq a \) (or \( x \neq b \)).

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

1. How do you solve a system of linear equations?
2. What are the properties of quadratic equations?
3. How do you find the roots of a polynomial equation?
4. What is the significance of discriminants in quadratic equations?
5. How do you solve equations involving absolute values?

**Tip:** When solving equations involving fractions, always look for a common denominator to simplify the equation before solving.

Learn how to solve equations involving fractions, specifically finding the value of x in the equation 1/(x-a) + 1/(x-b) = 2/(x-c). This detailed step-by-step solution covers fraction manipulation and cross-multiplication techniques. Suitable for advanced high school students.

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