To find the correct expression that should replace the word "numerator" in the provided mathematical work, let's carefully follow the steps and simplify the given expression:
Given the expression:
[tex]\[
\frac{2x - 8}{x^2 - 7x + 10} - \frac{x - 3}{x^2 - 7x + 10}
\][/tex]
Since the denominators are the same, we can combine the fractions by subtracting the numerators:
[tex]\[
\frac{(2x - 8) - (x - 3)}{x^2 - 7x + 10}
\][/tex]
Next, distribute the negative sign and simplify the numerator:
[tex]\[
(2x - 8) - (x - 3) = 2x - 8 - x + 3
\][/tex]
Combine like terms in the numerator:
[tex]\[
2x - x - 8 + 3 = x - 5
\][/tex]
So the simplified expression becomes:
[tex]\[
\frac{x - 5}{x^2 - 7x + 10}
\][/tex]
We also know that [tex]\( x^2 - 7x + 10 \)[/tex] can be factored as [tex]\((x-2)(x-5)\)[/tex]. With the numerator simplified, the final fraction is:
[tex]\[
\frac{x - 5}{(x-2)(x-5)}
\][/tex]
The expression that should replace the word "numerator" in the final simplified fraction is:
[tex]\[
x - 5
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{x - 5}
\][/tex]
