Question

what are the steps to this equasion
5/ x^2 -7x+12 - 2/x-3 = 5/x-4

Answer 1

To solve the equation:

\[
\frac{5}{x^2 - 7x + 12} - \frac{2}{x - 3} = \frac{5}{x - 4}
\]

we will follow these steps:

### Step 1: Factor the denominator
First, let's factor the quadratic in the first term's denominator:

\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\]

### Step 2: Rewrite the equation
Now we can rewrite the equation with the factored terms:

\[
\frac{5}{(x - 3)(x - 4)} - \frac{2}{x - 3} = \frac{5}{x - 4}
\]

### Step 3: Find a common denominator
The common denominator for all the fractions will be \((x - 3)(x - 4)\). Let's multiply each term by this common denominator to clear the fractions:

\[
5 - 2(x - 4) = 5(x - 3)
\]

### Step 4: Distribute and simplify
Now, distribute on the left side:

\[
5 - 2x + 8 = 5x - 15
\]

Combine like terms:

\[
13 - 2x = 5x - 15
\]

### Step 5: Solve for \(x\)
Now, move all terms involving \(x\) to one side and constant terms to the other:

\[
13 + 15 = 5x + 2x
\]
\[
28 = 7x
\]

Now, divide by \(7\):

\[
x = 4
\]

### Step 6: Check for extraneous solutions
We must check whether our solution \(x = 4\) is valid by substituting it back into the original equation. However, we notice that:

- \(x - 3\) is not equal to zero (since \(4 - 3 = 1\)),
- \(x - 4\) becomes zero (since \(4 - 4 = 0\)), hence the equation is undefined for this value.

### Conclusion
Since \(x=4\) makes the equation undefined, there are **no valid solutions** for the original equation.