Questions: Systems of Linear Equations Solving a (2 times 2) system of linear equations that is inconsistent or consistent... Two systems of equations are given below. For each system, choose the best description of its solution. If applicable, give the solution. System A [ beginarrayr x+2 y=8 -x-2 y=8 endarray ] 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 [ beginaligned x+3 y =9 -x =3 y-9 endaligned ] 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

1. System A:

   • Add the two equations to see if they result in a contradiction or a valid equation. If they result in a contradiction, the system has no solution. If they simplify to a valid equation, check for unique or infinite solutions.

2. System B:

   • Simplify the second equation and compare it with the first equation. If they are equivalent, the system has infinitely many solutions. If they are not equivalent, solve the system to find a unique solution or determine if there is no solution.

For System A, we have the equations: \[ \begin{aligned}

1. & \quad x + 2y = 8 \\
2. & \quad -x - 2y = 8 \end{aligned} \] Adding these two equations results in: \[ 0 = 16 \] This is a contradiction, indicating that System A has no solution.

For System B, we have the equations: \[ \begin{aligned}

1. & \quad x + 3y = 9 \\
2. & \quad -x = 3y - 9 \end{aligned} \] Rearranging the second equation gives: \[ x = 9 - 3y \] This shows that the second equation is dependent on the first. Therefore, System B has infinitely many solutions, which can be expressed as: \[ y = \frac{9 - x}{3} \]

Final Answer