Let's analyze the given statements mathematically.

Let the number be $N$, and the smaller number be $x$.

First condition:

$N = 2x + 3 \quad \text{(Equation 1)}$

Second condition:

$N = 2(x + 1) \quad \text{(Equation 2)}$

Simplify the second equation: $N = 2x + 2 \quad \text{(Equation 2 simplified)}$

Now, let's compare the two equations:

1. From Equation 1: $N = 2x + 3$
2. From Equation 2: $N = 2x + 2$

Analysis:

The two equations $N = 2x + 3$ and $N = 2x + 2$ have the same slope (both have a coefficient of $2x$), but they have different y-intercepts (3 and 2, respectively).

Since the lines are parallel and have different y-intercepts, they will never intersect. This implies that there is no solution for the system of equations.

Conclusion:

The correct answer is:

No solutions exist because the situation describes two lines that have the same slope and different y-intercepts.

Would you like further details or have any other questions?

Here are some related questions you might consider:

1. What happens if the lines have the same slope and the same y-intercept?
2. How do you determine if two lines are parallel?
3. Can two lines with different slopes ever be parallel?
4. What is the general form of a linear equation?
5. How do you find the point of intersection between two lines?
6. What is the significance of the y-intercept in a linear equation?
7. Can two lines with the same slope and different y-intercepts ever intersect?
8. How can you visually represent the relationship between two linear equations?

Tip: When analyzing systems of linear equations, always check both the slopes and y-intercepts to determine the number of solutions.