Question

Let x and y be integers. If x and y satisfy 37x + 82y = -172 and x + y = -1, then find x.

Answer 1

We are given two equations involving integers \( x \) and \( y \):

1. \( 37x + 82y = -172 \) \quad (1)
2. \( x + y = -1 \) \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

\[
\boxed{2}
\]