Question: The system has no solution. System A \[ -2x + y = -4 \\ -2x - y = -4 \] The system has a…

The system has no solution.

System A \[ -2x + y = -4 \\ -2x - y = -4 \]

The system has a unique solution: \[ (x, y) = (\boxed{\phantom{0}}, \boxed{\phantom{0}}) \]

The system has infinitely many solutions. They must satisfy the following equation: \[ y = \boxed{\phantom{0}} \]

Solution

Let’s solve the system of equations to determine the number of solutions. The given system is: \[ \begin{cases} -2x + y = -4 & \\ -2x - y = -4 & \end{cases} \] Add the two equations together to eliminate \(y\): \[ (-2x + y) + (-2x - y) = -4 + (-4) \] Simplifying gives: \[ -4x = -8 \] Solve for \(x\): \[ x = \frac{-8}{-4} = 2 \] Substitute \(x = 2\) back into the first equation: \[ -2(2) + y = -4 \] \[ -4 + y = -4 \] \[ y = 0 \] The solution to the system is \((x, y) = (2, 0)\). The system has a unique solution: \((x, y) = (2, 0)\).