Question

why equations 8v - 3u = 5uv and 6v -5 u = -2uv are not linear??????

Answer 1

8 v - 3 u = 5 u v      and   6v - 5 u = - 2 uv

These are not linear equations in variables u and v. In linear equations we have terms of degree 1 or 0 only.  We don't have a term with both variables.  The term with  uv  is treated as equivalent to a second degree term.

But these can be converted into linear equations....

Let  1/u = x    and  1/v = y... 
Divide the equations by  uv.

So   8/u - 3/v = 5       and   6/u - 5/v = - 2
or     8x - 3 y = 5                6x - 5 y + 2 = 0

These equations are linear as the terms are of degree 1 or 0 only.

Answer 2

8v -3u = 5uv --------(1)

6v -5u = -2uv ---------(2)

multiply 5 with first and 3 with second and then subtract .

5(8v -3u) -3(6v-5u) =5×5uv -3×(-2uv)

40v -15u -18v +15u = 25uv +6uv

22v = 31uv

u = 22/31

put this in equation (1)
8/u - 3/ v = 5

176/31 - 3/v = 5

(176 -155)/31 = 3/v

21/31 = 3/v

v = 31/7

why this equation not linear ??
8u -3v =5uv

if you divide , uv by both sides then,

8 /u - 3/v = 5

here you can see that term u and v are inversely related to each other .
but in linear ,
ax + by =c type equation should be exist . so , these equation are not linear .