Q:

I can't solve this Algebra 1 equation: For how long did she run?

Before going to school, Eudora ran from her home to a secret laboratory at an average speed of 12 km/h. Since she was running late, she then took one of her jetpacks and flew to her school at an average speed of 76 km/h. Eudora traveled a total distance of 120 kilometers, and the entire trip took 2 hours.
How long did Eudora spend running, and how long did she spend flying using her jetpack?
I have tried this so far:
$r$ = time spent running;
$f$ = time spent flying.
$r$ + $f$ = 2
$\frac{r}{12}$Km+$\frac{f}{76}$Km = 120km

In order to get rid of the 12, I multiplied:

(-12)($\frac{r}{12}$Km+$\frac{f}{76}$Km) = 120km(-12)

= -$r$ - $\frac{12}{76}f$ = -1440

I added the two equations:

$r$ + $f$ = 2

+

-$r$ - $\frac{12}{76}f$ = -1440$\frac{64}{76}f$=-1440
When I simplify this, I get a huge answer in the thousands. 
All of the equations that I have tried are wrong. I don't know what the answer is and I have been stuck on this one for a while now. What am I doing wrong and how can I fix it? 

A:

She ran for $R$ hours at 12 km/h.
She jetpacked for $J$ hours at 76 km/h.
We are told that 
$J+R=2$ (so $J=2-R$)
and that
$12R+76J=120$.
By substituting for $J$ we get:
$12R+76(2-R)=120$
$\Rightarrow 12R+152-76R=120$
$\Rightarrow -64R+152=120$
$\Rightarrow 32=64R$
$\Rightarrow R = \dfrac{1}{2}$.
$\therefore$ She ran for half an hour and jetpacked for one and half hours.

