**Solving Simultaneous Equations: 6x + 3y = 6xy and 2x + 4y = 5xy by Reduction Method**

In this article, we will learn how to solve a system of simultaneous equations using the reduction method. The two equations we will be working with are:

**Equation 1:** 6x + 3y = 6xy
**Equation 2:** 2x + 4y = 5xy

**Step 1: Multiply the Equations**

To eliminate one of the variables, we will multiply Equation 1 by 2 and Equation 2 by 3, which results in:

**New Equation 1:** 12x + 6y = 12xy
**New Equation 2:** 6x + 12y = 15xy

**Step 2: Subtract the Equations**

Now, we will subtract New Equation 2 from New Equation 1 to eliminate the x-term:

**(12x + 6y) - (6x + 12y) = (12xy) - (15xy)**
**6x - 6y = -3xy**

**Step 3: Simplify the Result**

We can simplify the resulting equation by dividing both sides by -6:

**x - y = xy/2**

**Step 4: Substitution**

Now that we have a simpler equation, we can substitute x - y = xy/2 back into one of the original equations. Let's use Equation 1:

**6x + 3y = 6xy**
**(x + y) + (x - y) = 6xy**
**(x + y) + (xy/2) = 6xy**

**Step 5: Solve for x and y**

Solving for x and y involves some algebraic manipulations. Let's start by solving for x + y:

**x + y = 10xy - xy/2**
**x + y = (20xy - xy)/2**
**x + y = 19xy/2**

Now, we can substitute x + y = 19xy/2 back into x - y = xy/2 to solve for x:

**x - y = xy/2**
**(19xy/2 - y) - y = xy/2**
**19xy/2 - 2y = xy/2**
**19xy - 4y = xy**
**18xy = 4y**
**y = 9x/2**

Substituting y = 9x/2 back into x - y = xy/2, we can solve for x:

**x - y = xy/2**
**x - 9x/2 = x(9x/2)/2**
**x/2 = 9x^2/4**
**x = 18/9**
**x = 2**

Now that we have x, we can find y:

**y = 9x/2**
**y = 9(2)/2**
**y = 9**

Therefore, the solution to the system of simultaneous equations is x = 2 and y = 9.