Questions: Two systems of equations are given below. For each system, choose the best description of its solution. If applicable, give the solution. System A x-2 y=6 -x+2 y=6 - The system has no solution. - The system has a unique solution: (x, y)= - The system has infinitely many solutions. They must satisfy the following equation: y= System B x-5 y =-5 -x+5 y =5 - The system has no solution. - The system has a unique solution: (x, y)=(, ) - The system has infinitely many solutions. They must satisfy the following equation: y=

Solution Steps

To determine the nature of the solutions for each system of equations, we can analyze the equations to see if they are consistent, inconsistent, or dependent. For System A, we can add the two equations to see if they result in a contradiction or a valid equation. For System B, we can do the same to determine if the system is consistent or not.

The equations for System A are: \[ \begin{aligned}

1. & \quad x - 2y = 6 \\
2. & \quad -x + 2y = 6 \end{aligned} \] Adding these two equations results in: \[ (x - 2y) + (-x + 2y) = 6 + 6 \implies 0 = 12 \] This is a contradiction, indicating that System A has no solution.

The equations for System B are: \[ \begin{aligned}

1. & \quad x - 5y = -5 \\
2. & \quad -x + 5y = 5 \end{aligned} \] Adding these two equations results in: \[ (x - 5y) + (-x + 5y) = -5 + 5 \implies 0 = 0 \] This indicates that the equations are dependent, leading to no unique solution. Therefore, System B also has no solution.

Final Answer