To solve the equation \( \frac{x}{3} + \frac{4}{y} = -1 \), we can follow these steps:

1. **Isolate one of the variables**: Let's isolate \( \frac{x}{3} \).

\[
\frac{x}{3} = -1 - \frac{4}{y}
\]

2. **Clear the fraction involving \( x \)**: Multiply both sides by 3 to solve for \( x \).

\[
x = 3 \left( -1 - \frac{4}{y} \right)
\]

\[
x = -3 - \frac{12}{y}
\]

So, \( x \) in terms of \( y \) is:

\[
x = -3 - \frac{12}{y}
\]

Alternatively, we can isolate \( \frac{4}{y} \):

\[
\frac{4}{y} = -1 - \frac{x}{3}
\]

3. **Clear the fraction involving \( y \)**: Multiply both sides by \( y \) to solve for \( y \).

\[
4 = y \left( -1 - \frac{x}{3} \right)
\]

\[
4 = -y - \frac{xy}{3}
\]

Multiply both sides by 3 to clear the fraction:

\[
12 = -3y - xy
\]

Rearrange to solve for \( y \):

\[
xy + 3y = -12
\]

\[
y(x + 3) = -12
\]

\[
y = \frac{-12}{x + 3}
\]

So, \( y \) in terms of \( x \) is:

\[
y = \frac{-12}{x + 3}
\]

These are the solutions for \( x \) in terms of \( y \) and \( y \) in terms of \( x \).