# Solve for x and y:x + y = 5xy, 3x + 2y = 13xy (x ≠ 0, y ≠ 0)

We have

x + y = 5xy

and 3x + 2y = 13xy

where x≠0 and y≠0

Lets simplify these equations.

x + y = 5xy

Dividing the equation by xy throughout,  Assuming p = 1/y and q = 1/x, we get

p + q = 5 …(i)

Also, 3x + 2y = 13xy

Dividing the equation by xy throughout,  Assuming p = 1/y and q = 1/x, we get

3p + 2q = 13 …(ii)

To solve these equations, we need to make one of the variables (in both the equations) have same coefficient.

Lets multiply eq.(i) by 2, so that variable q in both the equations have same coefficient.

Recalling equations (i) & (ii),

p + q = 5 [×2]

3p + 2q = 13 - p = - 3

p = 3

Substitute p = 3 in eq.(i)/eq.(ii), as per convenience of solving.

Thus, substituting in eq.(i), we get

3 + q = 5

q = 5 – 3

q = 2

Thus, p = 3 and q = 2

As q = 1/x,

2 = 1/x

x = 1/2

And p = 1/y

3 = 1/y

y = 1/3

Hence, we have x = 1/2 and y = 1/3

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