The equation \(3x + 4y = 5z\) is a linear equation in three variables: \(x\), \(y\), and \(z\). This type of equation represents a plane in three-dimensional space. To analyze or solve this equation, you typically need additional information or constraints. Here are a few ways you might proceed:

### Solving for One Variable
You can solve for one variable in terms of the other two. For example, solving for \(z\):

\[ z = \frac{3x + 4y}{5} \]

### Parametric Form
You can express two variables in terms of a parameter and then solve for the third. For example, let \(x = t\) and \(y = s\), then:

\[ z = \frac{3t + 4s}{5} \]

So, the parametric form is:

\[ (x, y, z) = (t, s, \frac{3t + 4s}{5}) \]

### Intersection with Coordinate Planes
You can find the intersections of the plane with the coordinate planes:

1. **Intersection with the \(xy\)-plane (\(z = 0\))**:
\[ 3x + 4y = 0 \]
This is a line in the \(xy\)-plane.

2. **Intersection with the \(xz\)-plane (\(y = 0\))**:
\[ 3x = 5z \]
\[ z = \frac{3x}{5} \]
This is a line in the \(xz\)-plane.

3. **Intersection with the \(yz\)-plane (\(x = 0\))**:
\[ 4y = 5z \]
\[ z = \frac{4y}{5} \]
This is a line in the \(yz\)-plane.

### Specific Solutions
If you have specific values for two of the variables, you can solve for the third. For example, if \(x = 1\) and \(y = 2\):

\[ 3(1) + 4(2) = 5z \]
\[ 3 + 8 = 5z \]
\[ 11 = 5z \]
\[ z = \frac{11}{5} \]

### Graphical Representation
Graphically, the equation represents a plane in three-dimensional space. You can plot this plane using software tools like MATLAB, Mathematica, or even graphing calculators that support 3D plotting.

### System of Equations
If you have additional equations involving \(x\), \(y\), and \(z\), you can solve the system of equations using methods like substitution, elimination, or matrix operations.

If you have any specific context or additional equations, please provide them, and I can give a more detailed solution or analysis.