Sure, let's solve for [tex]\( r \)[/tex] in the equation [tex]\( t = \frac{r}{r - 3} \)[/tex].
1. Start with the given equation:
[tex]\[
t = \frac{r}{r - 3}
\][/tex]
2. To eliminate the fraction, multiply both sides of the equation by [tex]\( r - 3 \)[/tex]:
[tex]\[
t(r - 3) = r
\][/tex]
3. Distribute [tex]\( t \)[/tex] on the left side:
[tex]\[
tr - 3t = r
\][/tex]
4. To isolate [tex]\( r \)[/tex], move all terms involving [tex]\( r \)[/tex] to one side of the equation:
[tex]\[
tr - r = 3t
\][/tex]
5. Factor out [tex]\( r \)[/tex] from the terms on the left side:
[tex]\[
r(t - 1) = 3t
\][/tex]
6. Finally, solve for [tex]\( r \)[/tex] by dividing both sides of the equation by [tex]\( t - 1 \)[/tex]:
[tex]\[
r = \frac{3t}{t - 1}
\][/tex]
So, the expression for [tex]\( r \)[/tex] in terms of [tex]\( t \)[/tex] is:
[tex]\[
r = \frac{3t}{t - 1}
\][/tex]
