Phoebe and Tucker are 48m apart. Phoebe looks up at an angle of 43 degrees to see a hot air balloon hovering in the sky directly between her and Tucker. Tucker looks up at an angle of 31 degrees to see the hot air balloon.

Make a sketch, with P for Phoebe and T for Tucker as a horizontal line, so that PT = 45 m
Let B be the position of the balloon, so that angle BPT = 43° and angle BTP = 31°
Let AB be the height of the balloon with A directly below B on PT

sin 43° = AB/BP ---> BP = AB/sin43
sin 31° = AB/BT ---> BT = AB/sin31
now we know, or should know, that sin 43>sin31
So AB divided by the larger number would produce the smaller answer.
So BP < BT
(notice I did not do any calculations)

To actually find PB, let PA = x, the AT = 48-x
tan 43 = AB/x
AB = xtan43

tan31 = AB/(48-x)
AB = (48-x)tan31

so xtan43 = 48-xtan31
xtan43 + tan31 = 48
x(tan43+tan31) = 48
x = 48/(tan43+tan31)

now cos 43 = x/BP
BP = x/cos43
= (48/(tan43 + tan31) )(1/cos43)
only now do we do any calculation ....

I get BP = 42.8 m

check my work please