Solving the Equation $\displaystyle \frac{x}{(x+y)} = \frac{7}{9}$

We are given two equations:

1. $3x - 7y = 14$
2. $\displaystyle \frac{x}{(x+y)} = \frac{7}{9}$

First, let's simplify the second equation to find a relationship between $x$ and $y$. Cross-multiplying gives:

$9x = 7(x+y)$

Distribute the 7:

$9x = 7x + 7y$

Subtract $7x$ from both sides:

$9x - 7x = 7y$

$2x = 7y$

We can express $y$ in terms of $x$:

$y = \frac{2}{7}x$

Substituting into $3x - 7y = 14$

Now, substitute the expression for $y$ from the second equation into the first equation:

$3x - 7\left(\frac{2}{7}x\right) = 14$

Simplify the term with $y$:

$3x - 2x = 14$

Combine the $x$ terms:

$x = 14$

Finding the Value of $y$

Use the relationship $y = \frac{2}{7}x$ and the value of $x$ we found:

$y = \frac{2}{7}(14)$

$y = 2 \times 2$

$y = 4$

Calculating the Final xy Value

Finally, calculate the value of $xy$ using the values $x=14$ and $y=4$:

$xy = (14)(4)$

$xy = 56$