
Use x, y, w, and z to clear up the confusion of RpTp and RcTc.
x = Rp
y = Tp
w = Rc
z = Tc
Therefore, you now need to solve for y and w.
xy = 288
wz = 108
x = 3w
y + z = 12
xy = 288
y = 288/x ← divided each side by x
wz = 108
z = 108/w ← divided each side by w
If y + z = 12, then
288/x + 108/w = 12 ← substituted for y and z
288w + 108x = 12xw ← multiplied each side by xw
288w + 108(3w) = 12w(3w) ← substituted 3w for x
288w + 324w = 36w² ← multiplied
612w = 36w² ← combined like terms
36w²  612w = 0 ← subtracted 612w from each side
36w(w  17) = 0 ← factored 36w out
36w = 0 or w  17 = 0 ← set each factor equal to zero
w = 0 or w = 17 ← solved for w
w = 17 only....w = 0 will not work later on
wz = 108 ← equation from above
17z = 108 ← substituted 17 for w
z = 6.353 ← divided by 17
y + z = 12 ← equation from above
y + 6.353 = 12 ← substituted 6.353 for z
y = 5.647 ← subtracted 6.353 from each side
w = 17 and y = 5.647
Rc = 17 and Tp = 5.647
 Proof
