Let's carefully analyze each given option to determine if any of them are equivalent to the original equation:
[tex]\[
\frac{15}{x} - \frac{3}{x-7}
\][/tex]
### Option a)
[tex]\( x(x-7) = 45 \)[/tex]
Analyzing this, we clearly see it does not represent a form involving the fractions [tex]\(\frac{15}{x}\)[/tex] and [tex]\(\frac{3}{x-7}\)[/tex]. Instead, it establishes a product equal to 45 which does not align with the structure of our original equation.
### Option b)
[tex]\( 15(x-7) = 3x \)[/tex]
Expanding this option:
[tex]\[
15x - 105 = 3x
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
15x - 3x = 105 \quad \Rightarrow \quad 12x = 105 \quad \Rightarrow \quad x = \frac{105}{12}
\][/tex]
This resulting equation is not in the same form as the given equation involving fractions.
### Option c)
[tex]\( 15x = 3(x - 7) \)[/tex]
Expanding this, we get:
[tex]\[
15x = 3x - 21
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
15x - 3x = -21 \quad \Rightarrow \quad 12x = -21 \quad \Rightarrow \quad x = -\frac{21}{12} = -\frac{7}{4}
\][/tex]
Again, this does not align with our initial equation's format with fractions.
### Option d)
[tex]\( 18 = x + (x + 7) \)[/tex]
Expanding this:
[tex]\[
18 = x + x + 7 \quad \Rightarrow \quad 18 = 2x + 7 \quad \Rightarrow \quad 2x = 11 \quad \Rightarrow \quad x = \frac{11}{2}
\][/tex]
This also does not match the original equation's structure.
### Conclusion
After carefully checking each of the options, none of them equate to the given equation:
[tex]\[
\frac{15}{x} - \frac{3}{x-7}
\][/tex]
None of the provided options accurately reflect the initial equation involving the fractions. Therefore, the correct answer is:
[tex]\[
\boxed{\text{None of the given options}}
\][/tex]
