To determine which of the following expressions is equivalent to [tex]\( r \)[/tex] in terms of [tex]\( S \)[/tex] and [tex]\( t \)[/tex] when [tex]\( S = \frac{r t - 3}{r - t} \)[/tex], we need to manipulate the initial equation to solve for [tex]\( r \)[/tex].
Given the equation:
[tex]\[ S = \frac{r t - 3}{r - t} \][/tex]
Our goal is to isolate [tex]\( r \)[/tex]. Let's proceed step-by-step:
1. Multiply both sides by [tex]\( r - t \)[/tex] to clear the denominator:
[tex]\[ S (r - t) = r t - 3 \][/tex]
2. Expand the left side of the equation:
[tex]\[ S r - S t = r t - 3 \][/tex]
3. Rearrange the terms to isolate those involving [tex]\( r \)[/tex] on one side:
[tex]\[ S r - r t = S t - 3 \][/tex]
4. Factor out [tex]\( r \)[/tex] on the left side:
[tex]\[ r (S - t) = S t - 3 \][/tex]
5. Solve for [tex]\( r \)[/tex] by dividing both sides of the equation by [tex]\( (S - t) \)[/tex]:
[tex]\[ r = \frac{S t - 3}{S - t} \][/tex]
Thus, the expression for [tex]\( r \)[/tex] in terms of [tex]\( S \)[/tex] and [tex]\( t \)[/tex] is:
[tex]\[ r = \frac{S t - 3}{S - t} \][/tex]
Now, we can compare this result to the given answer choices:
A. [tex]\( \frac{S t - 3}{S - t} \)[/tex]
B. [tex]\( \frac{S - 3}{S - 1} \)[/tex]
C. [tex]\( \frac{S - t}{S - 3} \)[/tex]
D. [tex]\( \frac{S t - 3}{S + t} \)[/tex]
E. [tex]\( \frac{3}{t - S} \)[/tex]
The correct equivalent expression for [tex]\( r \)[/tex] in terms of [tex]\( S \)[/tex] and [tex]\( t \)[/tex] is:
[tex]\[ \boxed{\frac{S t - 3}{S - t}} \][/tex]
So, the correct answer is [tex]\( \boxed{A} \)[/tex].
