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 \[ \begin{align*} -x + 3y &= 3 \\ x - 3y &= 3 \end{align*} \]

• 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 \[ \begin{align*} x - 3y &= -9 \\ -x + 3y &= 9 \end{align*} \]

• 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

Let’s analyze both systems of equations: System A: \[ \begin{cases} -x + 3y = 3 & \\ x - 3y = 3 & \end{cases} \] Step 1: Add the two equations together to eliminate \(x\): \[ (-x + 3y) + (x - 3y) = 3 + 3 \] This simplifies to: \[ 0 = 6 \] Since we obtained a contradiction, System A has no solution. System B: \[ \begin{cases} x - 3y = -9 & \\ -x + 3y = 9 & \end{cases} \] Step 1: Add the two equations together to eliminate \(x\): \[ (x - 3y) + (-x + 3y) = -9 + 9 \] This simplifies to: \[ 0 = 0 \] This indicates that the equations are dependent and represent the same line. Hence, System B has infinitely many solutions. To find the equation that solutions must satisfy, we can use either of the original equations from System B. Let’s take the first equation: \[ x - 3y = -9 \] Thus, the solutions must satisfy: \[ y = \frac{x + 9}{3} \]