Question: Two systems of equations are given below. For each system, choose the best description of its…

Two systems of equations are given below. For each system, choose the best description of its solution. If applicable, give the solution.

System A \[ -3x + y = -6 \\ 3x - y = 6 \]

• 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: \(y = \boxed{\phantom{0}}\)

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

• 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 systems of equations, we will use the method of elimination or substitution. System A: \[ \begin{cases} -3x + y = -6 & \\ 3x - y = 6 & \end{cases} \] Add the two equations: \[ (-3x + y) + (3x - y) = -6 + 6 \] This simplifies to: \[ 0 = 0 \] Since the result is a true statement, the system has infinitely many solutions. They must satisfy the equation \(y = 3x + 6\). System B: \[ \begin{cases} -5x + y = 5 & \\ 5x - y = 5 & \end{cases} \] Add the two equations: \[ (-5x + y) + (5x - y) = 5 + 5 \] This simplifies to: \[ 0 = 10 \] Since the result is a contradiction, the system has no solution.