Given system of equations:
1) 6x - 3y = 6xy
2) 2x - 4y = 5xy

To find the solution to the system of equations, we can use the method of substitution or elimination. Let's solve this system of equations step by step.

Step 1: Simplify the equations
We can simplify the equations by dividing both sides of each equation by the common factor of x.

1) 6 - 3y/x = 6y
2) 2 - 4y/x = 5y

Step 2: Set the equations equal to each other
We can set the simplified equations equal to each other to eliminate the x variable.

6 - 3y/x = 6y
2 - 4y/x = 5y

6 - 3y/x = 2 - 4y/x

Step 3: Multiply both sides by x
By multiplying both sides of the equation by x, we can eliminate the denominators.

x(6 - 3y/x) = x(2 - 4y/x)

6x - 3y = 2x - 4y

Step 4: Combine like terms
Combine the like terms on both sides of the equation.

6x - 2x - 3y + 4y = 0

4x + y = 0

Now, we have a new equation: 4x + y = 0

Step 5: Substitute y = -4x into one of the original equations
Substitute y = -4x into either equation 1) or 2) to find the values of x and y.

Let's substitute y = -4x into equation 1:

6x - 3(-4x) = 6x(-4x)

6x + 12x = -24x^2

18x = -24x^2

Step 6: Solve for x
Solve the quadratic equation to find the values of x.

18x = -24x^2

24x^2 + 18x = 0

6x(4x + 3) = 0

6x = 0 or 4x + 3 = 0

x = 0 or x = -3/4

Step 7: Substitute the values of x into y = -4x
Substitute the values of x into the equation y = -4x to find the corresponding values of y.

If x = 0, y = -4(0) = 0
If x = -3/4, y = -4(-3/4) = 3

Step 8: Solution
The solution to the system of equations is the set of values (x, y) that satisfy both equations. In this case, the solutions are:

(x, y) = (0, 0) and (x, y) = (-3/4, 3)

Conclusion
The given system of equations, 6x - 3y = 6xy and 2x - 4y = 5xy, has two solutions: (0, 0