What are solutions of simultaneous equations 1/x=(y-1)/y 3/x=(2y-1)/y

We'll re-write the given equations by cross
multiplying:

y = x(y-1)

We'll
remove the brackets:

y = xy -
x

We'll add x both sides:

x +
y = xy (1)

We'll change the 2nd equation
in:

3y = x(2y-1)

3y = 2xy -
x

We'll add x both sides:

3y +
x = 2xy (2)

We'll substitute (1) in
(2):

3y + x = 2(x+y)

3y + x =
2x + 2y

3y - 2y = 2x - x

y = x
(3)

We'll substitute (3) in
(2)

3x + x = 2x^2

4x =
2x^2

We'll divide by 2 and we'll use the symmetric
property:

x^2 - 2x = 0

We'll
factorize by x:

x(x-2) = 0

x =
0

x-2 = 0

x =
2

So, because x = y => y =
2

The solution of the system is {2 ;
2}.