Without working through anything, I notice a problem.
W=2X+4Y is a continuous randome variable, as well as the others.
So these sums cannot be one, but they can be approximately one.
This is a university paper question.
If X and Y are independent normal variables having a common mean μ such that
P(2X + 4Y <= 10) + P(3X + Y <=9) = 1 and
P(2X - 4Y <= 6) + P(Y - 3X >= 1) = 1
Determine the value of μ and the ratio of the variance of X and Y.