Answer:
Step-by-step explanation:
If we multiply x - 7 by x - 3, we get x^2 - 10x + 21. This shows that the LCD is x^2 - 10x + 21, or (x - 7)(x - 3). Let's convert each of the two fractions on the right to their equivalents, with the LCD in the denominator.
Multiplying both the numerator and the denominator of 6/(x - 7) by (x - 3), we get 6(x - 3) over x^2 - 10x + 21, which is the LCD.
Similarly, we get 5(x - 7) over x^2 - 10x + 21.
Now all three of the original fractions have the same denominator, x^2 - 10x + 21. We can now "drop" the LCD, which leaves us with:
x^2 - 29 = 6(x - 3) + 5(x - 7)
Performing the indicated multiplications, we get:
x^2 - 29 = 6x - 18 + 5x - 35. Combining like terms:
x^2 - 29 = 11x - 53, or x^2 - 11x + 24 = 0
This factors as follows: x^2 - 11x + 24 = 0 = (x - 3)(X - 8) = 0
Then the preliminary solutions are x = 3 and x = 8. Unfortunately, we cannot substitute x = 3 into the original equation, because that would lead to division by zero. The only valid solution is x = 8, since every term of the original equation is defined at x = 8, and the equation is true for this value.
Repeat: 3 cannot be a solution because the 3rd fraction is undefined at
x = 3. So we have only one valid solution (8) with no other solution to multiply by 8.
