Let's start with the given equation:
[tex]\[
\frac{1}{x} - \frac{1}{y} = \frac{1}{z}
\][/tex]
First, we need to find a common denominator for the fractions on the left-hand side:
[tex]\[
\frac{1}{x} - \frac{1}{y} = \frac{y}{xy} - \frac{x}{xy} = \frac{y - x}{xy}
\][/tex]
So the equation becomes:
[tex]\[
\frac{y - x}{xy} = \frac{1}{z}
\][/tex]
Next, we'll cross-multiply to solve for [tex]\( z \)[/tex]:
[tex]\[
z \cdot \frac{y - x}{xy} = 1
\][/tex]
Then, simplify this equation by eliminating the fraction:
[tex]\[
z \cdot (y - x) = xy
\][/tex]
Now, isolate [tex]\( z \)[/tex] on one side:
[tex]\[
z = \frac{xy}{y - x}
\][/tex]
So, based on the options provided, the value of [tex]\( z \)[/tex] is:
(d) [tex]\(\frac{xy}{y - x}\)[/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{d}
\][/tex]
