Question: System B \[ x - 5y = -5 \\ 5y = x + 5 \] The system has no solution. The system has a unique…

System B \[ x - 5y = -5 \\ 5y = x + 5 \]

The system has no solution. 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

To solve the given system of equations, we need to determine whether it has no solution, a unique solution, or infinitely many solutions. The system is: \[ \begin{cases} x - 5y = -5 & \\ 5y = x + 5 & \end{cases} \] First, let’s rewrite the second equation to match the form of the first equation. Solve for \(x\) in the second equation: \[ 5y = x + 5 \] Subtract 5 from both sides to write \(x\) in terms of \(y\): \[ x = 5y - 5 \] Now, substitute \(x = 5y - 5\) into the first equation: \[ (5y - 5) - 5y = -5 \] Simplify the equation: \[ 5y - 5 - 5y = -5 \] This reduces to: \[ -5 = -5 \] Since this equation is always true, it suggests that the original system has infinitely many solutions. The solutions must satisfy the equation: \[ y = y \] Therefore, the system has infinitely many solutions, and the solutions satisfy the relation already given in the system: \[ y = y \] So, the correct choice is that the system has infinitely many solutions.