To solve the equation \(\frac{1}{3}x + \frac{1}{4}y = 0\), we can follow these steps:

1. **Eliminate the fractions**: To make the equation easier to work with, let's eliminate the fractions by finding a common denominator. The common denominator for 3 and 4 is 12. Multiply every term in the equation by 12:

\[
12 \left(\frac{1}{3}x\right) + 12 \left(\frac{1}{4}y\right) = 12 \cdot 0
\]

2. **Simplify**: Perform the multiplications:

\[
4x + 3y = 0
\]

3. **Solve for one variable**: Let's solve for \(y\) in terms of \(x\):

\[
3y = -4x
\]

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

So, the solution to the equation \(\frac{1}{3}x + \frac{1}{4}y = 0\) is \(y = -\frac{4}{3}x\). This represents a line through the origin with a slope of \(-\frac{4}{3}\). Any pair \((x, y)\) that satisfies this relationship is a solution to the equation.