I'm having some issues with solving this general equilibrium exercise.
The way I started off is by assuming that since consumer 2's utility is fixed, he will have a fixed utility function. Then consumer 1 would shift his indifference curve to the point where they are tangent to maximise his own utility. Thus deriving the contract curve, by $$MRS^1=MRS^2$$ $$64x_1=117x_2-5x_1x_2$$ Then using the utility function: $$6=x{_{1}}^{1/2}x{_{2}}^{1/3}$$ I should be able to solve this because it's only 2 unknowns with 2 equations, but I cannot get anywhere without using an online calculator. I think the solution is allocation for consumer 1 is (4,8) and consumer 2: (9,8), but I am not sure how to get to this.
Can anyone find a similar exercise, where one consumer's utility was held fixed? Any advice on how to solve this?