To determine if the given system of equations has no solutions, infinitely many solutions, or exactly one solution, we need to analyze the two equations:

1. $-4x + 3y = -2$
2. $4x - y = -3$

Step 1: Rewrite both equations in standard form

The equations are already in standard form:

• Equation 1: $-4x + 3y = -2$
• Equation 2: $4x - y = -3$

Step 2: Analyze the system

We check whether the system is consistent, inconsistent, or dependent by comparing the slopes and constants.

Solve for the slope of each equation:

For the slope, rearrange each equation into slope-intercept form ($y = mx + b$).

• For $-4x + 3y = -2$: $3y = 4x - 2 \quad \Rightarrow \quad y = \frac{4}{3}x - \frac{2}{3}$ The slope is $m_1 = \frac{4}{3}$.

• For $4x - y = -3$: $-y = -4x - 3 \quad \Rightarrow \quad y = 4x + 3$ The slope is $m_2 = 4$.

Compare the slopes:

The slopes are different ($m_1 = \frac{4}{3}$ and $m_2 = 4$), meaning the lines are not parallel and therefore must intersect at exactly one point.

Step 3: Conclusion

The system has exactly one solution, as the lines are neither parallel nor overlapping.

Would you like a detailed calculation for solving the system to find the intersection point? Here are related questions to explore:

1. How can we verify the intersection point of these two lines?
2. What are the characteristics of a system with infinitely many solutions?
3. How do we recognize systems with no solutions geometrically?
4. Can substitution or elimination methods be used to solve this system?
5. What role do slopes play in determining the nature of solutions?

Tip: Always check slopes first to quickly assess if lines are parallel, intersecting, or overlapping!