Introduction:
In this problem, we are given two equations involving two variables u and v. Our task is to solve these equations and find the values of u and v.

Equations:
The given equations are:

4u-v=14uv

3u+2v=16uv

Method:
We can solve these equations using the method of substitution or elimination. Here, we will use the method of elimination to eliminate one variable and find the value of the other variable.

Solution:
Multiplying the second equation by 2, we get:

6u + 4v = 32uv

Now, subtracting the first equation from this, we get:

6u + 4v - 4u + v = 32uv - 14uv

Simplifying this, we get:

2u + 5v = 18uv

Rearranging this, we get:

2u = 18uv - 5v

Dividing both sides by 2v, we get:

u/v = (18 - 5v)/(2)

Similarly, multiplying the first equation by 3, we get:

12u - 3v = 42uv

Subtracting the second equation from this, we get:

12u - 3v - 3u - 2v = 42uv - 16uv

Simplifying this, we get:

9u - 5v = 26uv

Rearranging this, we get:

9u = 26uv + 5v

Dividing both sides by 9v, we get:

u/v = (26 + 5v)/(9)

Now we have two expressions for u/v. Equating them, we get:

(18 - 5v)/(2) = (26 + 5v)/(9)

Cross-multiplying and simplifying, we get:

81(18 - 5v) = 2(26 + 5v)

Expanding and simplifying, we get:

729 - 405v = 52 + 10v

Simplifying this, we get:

415v = 677

Dividing both sides by 415, we get:

v = 677/415

Substituting this value of v in one of the expressions for u/v, we get:

u/v = (18 - 5v)/(2) = (18 - 5(677/415))/(2)

Simplifying this, we get:

u/v = 23/83

Therefore, the solutions of the given equations are:

u = (23/83)v

v = 677/415

Conclusion:
We have solved the given equations using the method of elimination and found the values of u and v. The solutions are u = (23/83)v and v = 677/415.