In a strange alternate universe, pressure P, volume V , and temperature T are linked by the formula Pe T + Te V + ln 2 = 3P + V − 6. Suppose our current temperature is T = 0 and current pressure is P...

In a strange alternate universe, pressure P, volume V , and temperature T are linked by the formula

Pe^T
+ Te^V
+ ln 2 = 3P + V − 6.

Suppose our current temperature is T = 0 and current pressure is P = 3. Find the current volume V , then use linear approximation to estimate the total change in temperature ∆T if we change pressure by ∆P = 1 and change volume by ∆V = 2. Hint: Is there any way to get the pieces you need without ‘solving’ the awkward equation above?