Solve the equations

8v – 3u = 5uv

6v – 5u = -2uv

We have,

8v – 3u = 5uv …(i)

6v – 5u = -2uv …(ii)

In such case, divide the equations (i) and (ii) by uv, we get

From equation (i),

…(iii)

From equation (ii),

…(iv)

Substitute and in equations (iii) and (iv), we get

8x – 3y = 5 …(v)

6x – 5y = -2 …(vi)

In order to eliminate one of the variables (x and y), we need to make that variable’s coefficient equal in both of the equations.

In this question, we can eliminate y. Equation (v) has a (-) sign before y and equation (vi) too has a (-) sign before y.

Now, we need to make the coefficient of y equal, since in equation (v), y’s coefficient is 3 and in equation (vi), y’s coefficient is 5 (ignoring the signs before it).

For equal coefficient of y, multiply equation (v) by 5 and equation (vi) by 3. (Multiplication has to be done over the whole equation as to balance the equation even after making changes)

So, we have

8x – 3y = 5 [× 5

6x – 5y = -2 [× 3

⇒ 40x – 15y = 25 …(vii)

& 18x – 15y = -6 …(viii)

Now, we have equations (vii) and (viii) which can be solved by eliminating variable y.

Recall equation (vii) and (viii),

40x – 15y = 25

18x – 15y = -6

Solve these,

We get,

22x = 31

Put this values of x in equation (v), we get

⇒ 124 – 33y = 55

⇒ 33y = 124 – 55

⇒ 33y = 69

We know, and .

Or and .

And

Thus, the solution is and .