Answer
The number of rooms, bathrooms, fireplaces, and elevators are $132,35,28,3$ respectively.
Work Step by Step
Step 1. Assume the number of rooms, bathrooms, fireplaces, and elevators are $w,x,y,z$ respectively.
Step 2. Based on the given conditions, we have
$\begin{cases} w+x+y+z=198 \\ w-x-y=69 \\ y-z=25 \\ 2x-y-z=39 \end{cases}$
Step 3. Adding up the last two equations, we have $2x-2z=64$ or $x=z+32$ and the third equation gives $y=z+25$
Step 4. Substituting $x=z+32$ and $y=z+25$ in the first and second equations, we have
$\begin{cases} w+z+32+z+25+z=198 \\ w-z-32-z-25=69 \end{cases}$
or
$\begin{cases} w+3z=141 \\ w-2z=126 \end{cases}$
Step 5. Taking the difference of the two equations, we get $5z=15$ thus $z=3$
Step 6. Back-substitute to get $w=126+2(3)=132$, $x=35, y=28$
Step 7. Thus, the number of rooms, bathrooms, fireplaces, and elevators are $132,35,28,3$ respectively.
