If ` x - y = ( x + y)/(7) = (xy)/(4)` the numerical value of xy is

Options

A. `(4)/(3)`
B. `(3)/(4)`
C. `(1)/(4)`
D. `(1)/(3)`

Answer

To solve the equation \( x - y = \frac{x + y}{7} = \frac{xy}{4} \), we can follow these steps:

### Step 1: Set the expressions equal to a variable
Let \( k \) be the common value of the three expressions:
\[ x - y = k, \quad \frac{x + y}{7} = k, \quad \frac{xy}{4} = k \]

### Step 2: Express \( x + y \) and \( xy \) in terms of \( k \)
From the second equation, we can express \( x + y \):
\[ x + y = 7k \]
From the third equation, we can express \( xy \):
\[ xy = 4k \]

### Step 3: Solve for \( x \) and \( y \)
Now we have two equations:
1. \( x - y = k \) (Equation 1)
2. \( x + y = 7k \) (Equation 2)

We can add these two equations to eliminate \( y \):
\[ (x - y) + (x + y) = k + 7k \]
This simplifies to:
\[ 2x = 8k \implies x = 4k \]

Now, substitute \( x \) back into Equation 1 to find \( y \):
\[ 4k - y = k \implies y = 4k - k = 3k \]

### Step 4: Substitute \( x \) and \( y \) back into the expression for \( xy \)
Now we can find \( xy \):
\[ xy = (4k)(3k) = 12k^2 \]
But we also know from our earlier step that \( xy = 4k \). So we set these equal:
\[ 12k^2 = 4k \]

### Step 5: Solve for \( k \)
Divide both sides by \( k \) (assuming \( k \neq 0 \)):
\[ 12k = 4 \implies k = \frac{4}{12} = \frac{1}{3} \]

### Step 6: Find \( x \) and \( y \)
Now substitute \( k \) back to find \( x \) and \( y \):
\[ x = 4k = 4 \cdot \frac{1}{3} = \frac{4}{3} \]
\[ y = 3k = 3 \cdot \frac{1}{3} = 1 \]

### Step 7: Calculate \( xy \)
Now we can find \( xy \):
\[ xy = \left(\frac{4}{3}\right)(1) = \frac{4}{3} \]

Thus, the numerical value of \( xy \) is:
\[ \boxed{\frac{4}{3}} \]