If `2x + 6y = 3 xy and 10 x - 3y = 4 xy, ` find x,y

Options

A. `3,2`
B. `2,3`
C. `4,6`
D. `6,4`

Answer

To solve the equations \(2x + 6y = 3xy\) and \(10x - 3y = 4xy\), we will follow these steps:

### Step 1: Rewrite the equations
We have two equations:
1. \(2x + 6y = 3xy\) (Equation 1)
2. \(10x - 3y = 4xy\) (Equation 2)

### Step 2: Simplify Equation 1
We can divide the entire Equation 1 by 2 to simplify it:
\[ x + 3y = \frac{3}{2}xy \quad \text{(Equation 3)} \]

### Step 3: Rearrange Equation 3
Rearranging Equation 3 gives:
\[ x + 3y - \frac{3}{2}xy = 0 \]

### Step 4: Express \(x\) in terms of \(y\) from Equation 1
From Equation 3, we can express \(x\):
\[ x = \frac{3}{2}xy - 3y \]

### Step 5: Substitute \(x\) in Equation 2
Now, substitute \(x\) from Equation 3 into Equation 2:
\[ 10\left(\frac{3}{2}xy - 3y\right) - 3y = 4xy \]
Expanding this gives:
\[ 15xy - 30y - 3y = 4xy \]
\[ 15xy - 33y = 4xy \]

### Step 6: Rearrange to isolate terms
Rearranging gives:
\[ 15xy - 4xy = 33y \]
\[ 11xy = 33y \]

### Step 7: Solve for \(y\)
Now, we can divide both sides by \(y\) (assuming \(y \neq 0\)):
\[ 11x = 33 \]
\[ x = 3 \]

### Step 8: Substitute \(x\) back to find \(y\)
Now substitute \(x = 3\) back into Equation 1 or Equation 3 to find \(y\):
Using Equation 1:
\[ 2(3) + 6y = 3(3)y \]
\[ 6 + 6y = 9y \]
\[ 6 = 9y - 6y \]
\[ 6 = 3y \]
\[ y = 2 \]

### Final Answer
Thus, we find:
\[ x = 3, \quad y = 2 \]
So the solution is \((x, y) = (3, 2)\).

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