To determine the correct statements about the systems of equations, let's analyze both sets:

First System

$y = 6x - 3 \quad \text{and} \quad y = 3(2x + 1)$

1. Simplify the second equation:
   $y = 3(2x + 1) = 6x + 3$

2. The system becomes:
   $y = 6x - 3 \quad \text{and} \quad y = 6x + 3$

Here, the slopes ($6x$) are the same, but the $y$-intercepts are different ($-3$ and $+3$). Therefore, these two lines are parallel and never intersect.

Conclusion: This system has zero solutions.
Correct Statement:

"Zero Solutions: $y$ is equal to two different expressions."

Second System

$y = 2x + 1 \quad \text{and} \quad y = 6x + 3$

1. Compare the equations:
   $y = 2x + 1 \quad \text{and} \quad y = 6x + 3$

2. The slopes are different ($2$ and $6$), so the lines intersect at exactly one point.

Conclusion: This system has one solution.
Correct Statement:

"One Solution: $2x + 1 = 6x + 3$ has one solution."

Final Answers

1. First System: Zero Solutions: $y$ is equal to two different expressions.
2. Second System: One Solution: $2x + 1 = 6x + 3$ has one solution.

Let me know if you'd like step-by-step solving for intersection points or further explanations.

Related Questions

1. How do you determine if two lines are parallel or intersect?
2. What is the condition for infinitely many solutions in a system of equations?
3. How can you graphically represent two lines to identify their solution?
4. How do you solve a system of equations algebraically?
5. Can a system of nonlinear equations have zero, one, or infinitely many solutions?

Tip: When comparing equations, check the slopes and intercepts to identify parallel lines, intersecting lines, or coincident lines.