If ` 4x + 6y = 3 xy and 8x + 9y = 5xy ` then

Options

A. ` x = 2, y = 3 `
B. `x= 1, y = 2 `
C. ` x = 3, y = 4 `
D. `x = 1, y = - 1 `

Explanation

To solve the equations \( 4x + 6y = 3xy \) and \( 8x + 9y = 5xy \), we will follow these steps:

### Step 1: Rewrite the equations
We start with the two equations:
1. \( 4x + 6y = 3xy \)
2. \( 8x + 9y = 5xy \)

### Step 2: Divide both equations by \( xy \)
Dividing the first equation by \( xy \):
\[ \frac{4x}{xy} + \frac{6y}{xy} = \frac{3xy}{xy} \]
This simplifies to:
\[ \frac{4}{y} + \frac{6}{x} = 3 \quad \text{(Equation 1)} \]

Now, divide the second equation by \( xy \):
\[ \frac{8x}{xy} + \frac{9y}{xy} = \frac{5xy}{xy} \]
This simplifies to:
\[ \frac{8}{y} + \frac{9}{x} = 5 \quad \text{(Equation 2)} \]

### Step 3: Substitute variables
Let \( u = \frac{1}{x} \) and \( v = \frac{1}{y} \). Then, we can rewrite the equations:
1. \( 4v + 6u = 3 \)
2. \( 8v + 9u = 5 \)

### Step 4: Multiply the first equation by 2
To eliminate one variable, we can multiply the first equation by 2:
\[ 2(4v + 6u) = 2(3) \]
This gives us:
\[ 8v + 12u = 6 \quad \text{(Equation 3)} \]

### Step 5: Subtract Equation 2 from Equation 3
Now, we subtract Equation 2 from Equation 3:
\[ (8v + 12u) - (8v + 9u) = 6 - 5 \]
This simplifies to:
\[ 3u = 1 \]

### Step 6: Solve for \( u \)
Dividing both sides by 3 gives:
\[ u = \frac{1}{3} \]

### Step 7: Find \( x \)
Since \( u = \frac{1}{x} \), we have:
\[ \frac{1}{x} = \frac{1}{3} \implies x = 3 \]

### Step 8: Substitute \( x \) back to find \( y \)
Now, substitute \( x = 3 \) into Equation 1:
\[ 4v + 6\left(\frac{1}{3}\right) = 3 \]
This simplifies to:
\[ 4v + 2 = 3 \]
Subtracting 2 from both sides gives:
\[ 4v = 1 \implies v = \frac{1}{4} \]

### Step 9: Find \( y \)
Since \( v = \frac{1}{y} \), we have:
\[ \frac{1}{y} = \frac{1}{4} \implies y = 4 \]

### Final Answer
Thus, the values of \( x \) and \( y \) are:
\[ x = 3, \quad y = 4 \]