If `2/x+3/y=2` and `6/x+18/y=9`, then the values of x and y respectively are:

Options

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

Answer

To solve the equations \( \frac{2}{x} + \frac{3}{y} = 2 \) and \( \frac{6}{x} + \frac{18}{y} = 9 \), we will follow these steps:

### Step 1: Rewrite the equations
Let \( m = \frac{1}{x} \) and \( n = \frac{1}{y} \). Then we can rewrite the equations as:
1. \( 2m + 3n = 2 \) (Equation 1)
2. \( 6m + 18n = 9 \) (Equation 2)

### Step 2: Simplify Equation 2
We can simplify Equation 2 by dividing all terms by 3:
\[ 2m + 6n = 3 \]
Now we have:
1. \( 2m + 3n = 2 \) (Equation 1)
2. \( 2m + 6n = 3 \) (Equation 2)

### Step 3: Solve for one variable
Now, we can subtract Equation 1 from Equation 2:
\[ (2m + 6n) - (2m + 3n) = 3 - 2 \]
This simplifies to:
\[ 3n = 1 \]
Thus, we find:
\[ n = \frac{1}{3} \]

### Step 4: Substitute n back to find m
Now substitute \( n \) back into Equation 1 to find \( m \):
\[ 2m + 3\left(\frac{1}{3}\right) = 2 \]
This simplifies to:
\[ 2m + 1 = 2 \]
Subtracting 1 from both sides gives:
\[ 2m = 1 \]
Thus, we find:
\[ m = \frac{1}{2} \]

### Step 5: Find x and y
Recall that \( m = \frac{1}{x} \) and \( n = \frac{1}{y} \). Therefore:
\[ x = \frac{1}{m} = \frac{1}{\frac{1}{2}} = 2 \]
\[ y = \frac{1}{n} = \frac{1}{\frac{1}{3}} = 3 \]

### Final Answer
Thus, the values of \( x \) and \( y \) are:
\[ x = 2, \quad y = 3 \]