Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. \[ \begin{array}{r}-3 x+4 y=-5 \\ -6 x+8 y=-10\end{array} \] Answer One Solution No Solutions

The given system of equations is: \[ \begin{cases} -3x + 4y = -5 \\ -6x + 8y = -10 \end{cases} \] Let's analyze the system: 1. **Check for Proportionality:** - Multiply the first equation by 2: \[ 2(-3x + 4y) = 2(-5) \implies -6x + 8y = -10 \] - This result is exactly the same as the second equation. 2. **Interpretation:** - Since the second equation is a multiple of the first, both equations represent the same line. This means that every solution to the first equation is also a solution to the second equation. 3. **Conclusion:** - There are **infinitely many solutions** because the two equations describe the same line. **Answer:** Infinitely many solutions.