Let's solve the given systems of equations step by step.

System A:

Equations:

1. $-2x + y = 2$
2. $2x - y = 2$

Step 1: Add the equations to eliminate $y$: $(-2x + y) + (2x - y) = 2 + 2$ $0 = 4$

This is a contradiction, meaning System A has no solution.

System B:

Equations:

1. $-4x + y + 8 = 0$, simplify to: $-4x + y = -8$
2. $4x - y = 8$

Step 1: Add the equations to eliminate $y$: $(-4x + y) + (4x - y) = -8 + 8$ $0 = 0$

This is true, so the system might have infinitely many solutions. Let's confirm.

Step 2: Check if the equations are dependent:

From the first equation: $y = 4x - 8$

Substitute $y = 4x - 8$ into the second equation: $4x - (4x - 8) = 8$ $4x - 4x + 8 = 8$ $8 = 8$

This is consistent, so System B has infinitely many solutions. These solutions satisfy the equation: $y = 4x - 8$

Final Answers:

• System A: No solution.
• System B: Infinitely many solutions, $y = 4x - 8$.

Would you like me to clarify any part or explore further details?
Here are 5 related questions for deeper exploration:

1. How can we graphically interpret "no solution" for System A?
2. What does "infinitely many solutions" imply in terms of linear dependence?
3. Can we write the general solution for System B in parametric form?
4. How can substitution help us verify solutions to such systems?
5. Why does adding equations help simplify systems?

Tip: Always check for contradictions or dependencies when solving systems of equations!