Understanding the System of Equations
To analyze the given system of equations:
1. **Equations**:
- Equation 1: \(-3X + 4Y = 5\)
- Equation 2: \(\frac{9}{2}X - 6Y + \frac{15}{2}A = 0\)
2. **Identifying Variables**:
- We have three variables: \(X\), \(Y\), and \(A\).
- The equations involve two variables \(X\) and \(Y\) in the first equation and \(X\) and \(Y\) again, along with an additional variable \(A\) in the second equation.

Steps to Solve the System
- **Rearranging the First Equation**:
- Solve for \(Y\) in terms of \(X\):
- \(4Y = 3X + 5\)
- \(Y = \frac{3}{4}X + \frac{5}{4}\)
- **Substituting in the Second Equation**:
- Substitute \(Y\) from Equation 1 into Equation 2:
- \(\frac{9}{2}X - 6\left(\frac{3}{4}X + \frac{5}{4}\right) + \frac{15}{2}A = 0\)
- **Simplifying**:
- This leads to a linear equation in \(X\) and \(A\), which can be solved for one variable in terms of the other.

Determining the Nature of Solutions
- **Types of Solutions**:
- If the two equations are independent, there will be a unique solution.
- If they are dependent, the solution set will be infinite.
- If they are inconsistent, there will be no solution.

Conclusion
- The system of equations can yield:
- A unique solution if the equations intersect at one point.
- No solution if the lines are parallel.
- Infinite solutions if the equations represent the same line.
Through this analysis, we can conclude the behavior of the system based on the relationships between the equations.