To solve the system of equations using elimination, we have the following equations:
- \( 7x + 3y = -25 \)
- \( -7x + 9y = -47 \)
We can eliminate \( x \) by adding the two equations together since the coefficients of \( x \) are opposites:
\[ (7x + 3y) + (-7x + 9y) = -25 + (-47) \]
This simplifies to:
\[ (7x - 7x) + (3y + 9y) = -25 - 47 \]
\[ 0 + 12y = -72 \]
Dividing both sides by 12, we get:
\[ y = -6 \]
Now that we have the value of \( y \), we can substitute it back into one of the original equations to solve for \( x \). We'll use the first equation:
\[ 7x + 3(-6) = -25 \]
This simplifies to:
\[ 7x - 18 = -25 \]
Adding 18 to both sides:
\[ 7x = -25 + 18 \]
\[ 7x = -7 \]
Dividing both sides by 7, we get:
\[ x = -1 \]
Now we can express the solution as an ordered pair:
\[ (x, y) = (-1, -6) \]
Thus, the solution to the system of equations is:
\[ \boxed{(-1, -6)} \]