To determine the correct number of solutions for each system of equations, let’s analyze each system.

1. $3x - y = 4$ and $6x - 2y = 8$

By inspecting the second equation, $6x - 2y = 8$, we can see that it is equivalent to multiplying the first equation by 2: $2 \cdot (3x - y) = 2 \cdot 4 \Rightarrow 6x - 2y = 8$ Since the two equations are essentially the same, they are dependent and therefore have infinitely many solutions.

2. $-3x + y = 7$ and $2x - 4y = -8$

Let's try to manipulate the equations to see if we can find a solution or determine if they are inconsistent. Multiplying the first equation by 2 to align the coefficients of $y$: $-6x + 2y = 14$ However, the second equation is $2x - 4y = -8$.

These two lines have different slopes, so they intersect at a single point, meaning this system has one solution.

3. $y = -4x - 5$ and $y = -4x + 1$

Both equations have the same slope ($-4x$) but different y-intercepts ($-5$ and $1$). This implies that the lines are parallel and distinct, so they will never intersect. Therefore, this system has no solution.

Final Classification:

• Infinitely many solutions: $3x - y = 4$ and $6x - 2y = 8$
• One solution: $-3x + y = 7$ and $2x - 4y = -8$
• No solution: $y = -4x - 5$ and $y = -4x + 1$

Would you like further details or clarification on any part of this explanation?

Here are some related questions:

1. What are the different methods for solving systems of linear equations?
2. How can we determine the number of solutions of a system by looking at slopes and intercepts?
3. What is the geometric interpretation of systems with infinitely many solutions?
4. How do parallel lines influence the solution of a system of equations?
5. Can a system of equations have exactly two solutions?

Tip: When analyzing systems of equations, always check for proportional relationships between terms, as they often indicate dependency or parallelism.