Let's analyze the systems of equations given in the image and solve them step by step.

System A:

Equations:

1. $3x - y = 6$
2. $-3x + y = 6$

Add the two equations:

$(3x - y) + (-3x + y) = 6 + 6$ $0 = 12$

Since the result is a contradiction ($0 \neq 12$), System A has no solution. This means the lines are parallel and do not intersect.

System B:

Equations:

1. $-5x + y = -5$
2. $5x - y = 5$

Add the two equations:

$(-5x + y) + (5x - y) = -5 + 5$ $0 = 0$

This is a consistent equation. Let’s simplify both equations to check if they represent the same line:

1. Rewrite $-5x + y = -5$ as $y = 5x - 5$.
2. Rewrite $5x - y = 5$ as $y = 5x - 5$.

Both equations simplify to $y = 5x - 5$, meaning they represent the same line.

Thus, System B has infinitely many solutions, and the solutions must satisfy the equation: $y = 5x - 5$

Final Answers:

• System A: The system has no solution.
• System B: The system has infinitely many solutions, and the solutions must satisfy $y = 5x - 5$.

Do you need further details or explanations?

Related Questions:

1. How can you determine if two lines are parallel using their slopes?
2. What does it mean geometrically when a system of equations has no solution?
3. Can a system of linear equations have exactly two solutions? Why or why not?
4. How do you verify if two lines are identical algebraically?
5. What are the possible solution types for a system of linear equations in two variables?

Tip:

When solving systems of equations, always simplify the equations and compare slopes or directly eliminate variables to identify consistency.