Questions: System A -x+4 y =-8 x =8+4 y 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 the given system of equations, we need to analyze the equations:

1. Check if the equations are equivalent, which would imply infinitely many solutions.
2. Check if the equations are inconsistent, which would imply no solution.
3. If neither of the above, solve the system to find a unique solution.

We have the following system of equations: \[ \begin{aligned}

1. & \quad -x + 4y = -8 \\
2. & \quad x = 4y + 8 \end{aligned} \] To determine the nature of the solutions, we can express one variable in terms of the other.

From the second equation, we can express \( x \) in terms of \( y \): \[ x = 4y + 8 \]

Substituting \( x = 4y + 8 \) into the first equation: \[ -(4y + 8) + 4y = -8 \] This simplifies to: \[ -4y - 8 + 4y = -8 \] which is always true, indicating that the two equations are equivalent.

Since the equations are equivalent, the system has infinitely many solutions. These solutions can be expressed in terms of \( y \): \[ y = y \quad \text{(where \( y \) can take any real value)} \] Thus, we can express \( y \) in terms of \( x \): \[ y = \frac{x - 8}{4} \]

Final Answer