We are given two equations involving integers \( x \) and \( y \):
- \( 37x + 82y = -172 \) \quad (1)
- \( x + y = -1 \) \quad (2)
We can express \( y \) in terms of \( x \) from equation (2):
\[ y = -1 - x \]
Next, we will substitute this expression for \( y \) into equation (1):
\[ 37x + 82(-1 - x) = -172 \]
Expanding the left side:
\[ 37x - 82 - 82x = -172 \]
Combining like terms:
\[ (37x - 82x) - 82 = -172 \] \[ -45x - 82 = -172 \]
Now, we add 82 to both sides:
\[ -45x = -172 + 82 \] \[ -45x = -90 \]
Dividing by -45:
\[ x = \frac{-90}{-45} = 2 \]
Now, we have found \( x = 2 \). To find \( y \), we substitute \( x \) back into equation (2):
\[ y = -1 - 2 = -3 \]
Thus, we have \( x = 2 \) and \( y = -3 \).
Finally, we verify our solution by substituting \( x \) and \( y \) back into equation (1):
\[ 37(2) + 82(-3) = 74 - 246 = -172 \]
Both equations are satisfied.
Therefore, the value of \( x \) is
\[ \boxed{2} \]