Question

Solve for x and y:

x + y = 5xy, 3x + 2y = 13xy (x ≠ 0, y ≠ 0)

Answer 1

The given equations are:

x + y = 5xy   ...(i)

3x + 2y = 13xy   ...(ii)

From equation (i), we have:

`(x + y)/(xy) = 5`

⇒ `1/y + 1/x = 5`   ...(iii)

For equation (ii), we have:

`(3x + 2y)/(xy) = 13`

⇒ `3/y + 2/x = 13`   ...(iv)

On substituting `1/y = v` and `1/x = u`, we get:

v + u = 5   ...(v)

3v + 2u = 13   ...(vi)

On multiplying (v) by 2, we get:

2v + 2u = 10   ...(vii)

On subtracting (vii) from (vi), we get:

v = 3

⇒ `1/y = 3` 

⇒ `y = 1/3`

On substituting `y = 1/3` in (iii), we get:

`1/(1/3) + 1/x = 5`

⇒ `3 + 1/x = 5`

⇒ `1/x = 2`

⇒ `x = 1/2`

Hence, the required solution is `x = 1/2` and `y = 1/3` or x = 0 and y = 0.