To solve this equation, we'll need to find the common denominator for all the fractions and then simplify the equation. Let's break it down step by step.
Step 1: Find the common denominator.
The denominators in this equation are x^2 - 7x + 12, x - 3, and x - 4. To find the common denominator, we need to factor each denominator and see what factors they have in common.
The factors of x^2 - 7x + 12 are (x - 3) and (x - 4).
The factor of x - 3 is (x - 3).
The factor of x - 4 is (x - 4).
Since all three denominators share the factors (x - 3) and (x - 4), the common denominator is (x - 3)(x - 4).
Step 2: Rewrite the fractions with the common denominator.
Now that we have the common denominator, we need to rewrite each fraction in terms of this denominator. To do this, we multiply the numerator and the denominator of each fraction by the factors that are missing in their denominators.
The first fraction is 5/(x^2 - 7x + 12). To rewrite this fraction with the common denominator, we multiply both the numerator and the denominator by (x - 3)(x - 4), which gives us (5(x - 3)(x - 4))/((x - 3)(x - 4)).
The second fraction is 2/(x - 3). To rewrite this fraction with the common denominator, we multiply both the numerator and the denominator by (x - 4), which gives us (2(x - 4))/((x - 3)(x - 4)).
Step 3: Simplify the equation.
Now that we have rewritten all the fractions with the common denominator, we can combine them into a single fraction and simplify the equation.
Combining the fractions, we have:
(5(x - 3)(x - 4) + 2(x - 4))/((x - 3)(x - 4)) = 5/(x - 4).
Expanding and simplifying the numerator, we get:
(5x^2 - 38x + 60 + 2x - 8)/((x - 3)(x - 4)) = 5/(x - 4).
Combining like terms in the numerator, we have:
(5x^2 - 36x + 52)/((x - 3)(x - 4)) = 5/(x - 4).
We can now cancel out the common factors in the numerator and denominator:
5x^2 - 36x + 52 = 5(x - 3).
Expanding the right side, we get:
5x^2 - 36x + 52 = 5x - 15.
Step 4: Solve the equation.
Now that we have a simplified equation, we can solve it by moving all terms to one side and setting the equation equal to zero.
Subtracting 5x and 52 from both sides, we get:
5x^2 - 41x - 67 = 0.
This is now a quadratic equation. To solve it, you can use factoring, completing the square, or the quadratic formula.