System of Equations
The given system of equations is:
1. \(-3x + 4y = 5\)
2. \(\frac{9}{2}x - 6y + \frac{15}{2} = 0\)

Step 1: Simplifying the Second Equation
To analyze the second equation, we can simplify it:
- Rearranging:
\(\frac{9}{2}x - 6y = -\frac{15}{2}\)
- Multiplying through by 2 to eliminate fractions:
\(9x - 12y = -15\)
Now, we have the simplified system:
1. \(-3x + 4y = 5\)
2. \(9x - 12y = -15\)

Step 2: Solving the System
To determine the relationship between the two equations, we can manipulate them further.
- From the first equation, isolate \(y\):
\(4y = 3x + 5\)
\(y = \frac{3}{4}x + \frac{5}{4}\)
- From the second equation, isolate \(y\):
\(12y = 9x + 15\)
\(y = \frac{3}{4}x + \frac{5}{4}\)

Step 3: Conclusion
Both equations rearrange to the same linear function:
- \(y = \frac{3}{4}x + \frac{5}{4}\)
This indicates that the two equations represent the same line.

Result
- The system has **infinitely many solutions** since both equations are equivalent.
- Any point on the line defined by \(y = \frac{3}{4}x + \frac{5}{4}\) is a solution to the system.