To find the inverse of Janine's equation, we need to solve for [tex]\( r \)[/tex] in terms of [tex]\( t \)[/tex] from the given equation.
First, we start with the original equation:
[tex]\[ t = \frac{120r}{r + 120} \][/tex]
We want to isolate [tex]\( r \)[/tex]. To do that, let's follow these steps:
1. Multiply through by [tex]\( r + 120 \)[/tex] to clear the fraction:
[tex]\[ t(r + 120) = 120r \][/tex]
2. Distribute the [tex]\( t \)[/tex] on the left-hand side:
[tex]\[ tr + 120t = 120r \][/tex]
3. Rearrange the equation to group terms involving [tex]\( r \)[/tex]:
[tex]\[ tr - 120r = -120t \][/tex]
4. Factor out [tex]\( r \)[/tex] on the left side:
[tex]\[ r(t - 120) = -120t \][/tex]
5. Solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{-120t}{t - 120} \][/tex]
Now we have expressed [tex]\( r \)[/tex] in terms of [tex]\( t \)[/tex]. Thus, the inverse of Janine's equation is:
[tex]\[ r = \frac{-120t}{t - 120} \][/tex]
This is the required inverse equation.
