To solve this equation, we first need to find common denominators for all the fractions:
(2x - 7)/(x^2 - 9x + 14) - 1/(x^2 - 3x + 2) = 1/(x - 1)
The denominators are already factored, so the common denominator is (x - 7)(x - 2)(x - 1).
Next, rewrite the fractions with the common denominator:
[(2x - 7)(x - 2)]/[(x - 7)(x - 2)(x - 1)] - [(x - 7)(x - 1)]/[(x - 7)(x - 2)(x - 1)] = 1/(x - 1)
Multiply each term by the common denominator:
(2x - 7)(x - 2) - (x - 7)(x - 1) = (x - 7)(x - 2)
Expand and simplify each side:
2x^2 - 11x + 14 - x^2 + 8x - 7 = x^2 - 2x - 7
x^2 - 3x + 7 = x^2 - 2x - 7
Combine like terms:
-3x + 7 = -2x - 7
Add 2x to both sides:
-x + 7 = -7
Subtract 7 from both sides:
-x = -14
Multiply both sides by -1 to solve for x:
x = 14
Therefore, the solution to the equation is x = 14.
To solve this equation, we first need to find common denominators for all the fractions:
(2x - 7)/(x^2 - 9x + 14) - 1/(x^2 - 3x + 2) = 1/(x - 1)
The denominators are already factored, so the common denominator is (x - 7)(x - 2)(x - 1).
Next, rewrite the fractions with the common denominator:
[(2x - 7)(x - 2)]/[(x - 7)(x - 2)(x - 1)] - [(x - 7)(x - 1)]/[(x - 7)(x - 2)(x - 1)] = 1/(x - 1)
Multiply each term by the common denominator:
(2x - 7)(x - 2) - (x - 7)(x - 1) = (x - 7)(x - 2)
Expand and simplify each side:
2x^2 - 11x + 14 - x^2 + 8x - 7 = x^2 - 2x - 7
x^2 - 3x + 7 = x^2 - 2x - 7
Combine like terms:
-3x + 7 = -2x - 7
Add 2x to both sides:
-x + 7 = -7
Subtract 7 from both sides:
-x = -14
Multiply both sides by -1 to solve for x:
x = 14
Therefore, the solution to the equation is x = 14.