If `2/x+3/y=9/(xy) and 4/x+9/y=21/(xy)` where, `x ne 0,y ne 0, ` then what is the value of x + y?

Options

A. 2
B. 3
C. 8
D. 4

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \( \frac{2}{x} + \frac{3}{y} = \frac{9}{xy} \)
2. \( \frac{4}{x} + \frac{9}{y} = \frac{21}{xy} \)

### Step 2: Multiply through by \( xy \)
To eliminate the fractions, we can multiply both sides of each equation by \( xy \).

For the first equation:
\[ xy \left( \frac{2}{x} + \frac{3}{y} \right) = xy \cdot \frac{9}{xy} \]
This simplifies to:
\[ 2y + 3x = 9 \quad \text{(Equation 1)} \]

For the second equation:
\[ xy \left( \frac{4}{x} + \frac{9}{y} \right) = xy \cdot \frac{21}{xy} \]
This simplifies to:
\[ 4y + 9x = 21 \quad \text{(Equation 2)} \]

### Step 3: Rearranging the equations
Now we have two linear equations:
1. \( 2y + 3x = 9 \)
2. \( 4y + 9x = 21 \)

### Step 4: Solve for one variable
From Equation 1, we can express \( y \) in terms of \( x \):
\[ 2y = 9 - 3x \implies y = \frac{9 - 3x}{2} \]

### Step 5: Substitute \( y \) into Equation 2
Now substitute \( y \) into Equation 2:
\[ 4\left(\frac{9 - 3x}{2}\right) + 9x = 21 \]
This simplifies to:
\[ 2(9 - 3x) + 9x = 21 \]
\[ 18 - 6x + 9x = 21 \]
\[ 18 + 3x = 21 \]
\[ 3x = 21 - 18 \]
\[ 3x = 3 \implies x = 1 \]

### Step 6: Find \( y \)
Now substitute \( x = 1 \) back into the equation for \( y \):
\[ y = \frac{9 - 3(1)}{2} = \frac{9 - 3}{2} = \frac{6}{2} = 3 \]

### Step 7: Find \( x + y \)
Now we can find \( x + y \):
\[ x + y = 1 + 3 = 4 \]

### Final Answer
The value of \( x + y \) is \( \boxed{4} \).