To determine the number of solutions for the system of equations, we need to check if the lines represented by each equation intersect, are parallel, or are coincident.

The system of equations is:

1. $y = -\frac{5}{6}x + 8$
2. $y = \frac{10}{7}x - \frac{7}{5}$

To analyze the number of solutions, let's compare the slopes and y-intercepts of these equations:

1. For the first equation, the slope is $-\frac{5}{6}$ and the y-intercept is 8.
2. For the second equation, the slope is $\frac{10}{7}$ and the y-intercept is $-\frac{7}{5}$.

Analysis

1. Different Slopes: Since the slopes of the two lines ($-\frac{5}{6}$ and $\frac{10}{7}$) are different, the lines are not parallel. Therefore, they will intersect at a single point.
2. Single Intersection Point: Since they intersect at only one point, there is exactly one solution to this system of equations.

Conclusion

The system of equations has one solution.

Would you like a more detailed explanation on how to verify this solution?

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Here are five related questions to expand your understanding:

1. How do you determine if two lines are parallel or perpendicular based on their slopes?
2. What does it mean geometrically if two lines in a system of equations have the same slope and y-intercept?
3. How can you find the intersection point of two lines algebraically?
4. How can systems of equations be classified as consistent, inconsistent, or dependent?
5. What are some real-world applications of solving systems of equations?

Tip: In systems of equations, checking the slopes first can quickly tell you if the lines will intersect, which is a fast way to determine if there's a single solution.