 If (2,3) and (-3,5) are both solutions to a system of two linear equations, does the system have any other solutions?  Explain.  

If the points \((2,3)\) and \((-3,5)\) are both solutions to a system of two linear equations, this suggests that these points lie on both lines represented by the equations in the system.

For two linear equations to have exactly these two solutions and no others, the equations would need to represent two lines that intersect at only these two points. However, this isn't possible with straight lines in a 2D coordinate plane. A linear equation describes a straight line, and any two straight lines in the plane can either:

1. Be the same line (in which case they have infinitely many solutions, as all points on the line are solutions).
2. Be parallel with no solutions (if they do not intersect).
3. Intersect at exactly one point (one unique solution).

Since the points \((2,3)\) and \((-3,5)\) are both solutions, the two lines must coincide, meaning they are the same line. Therefore, the system of equations would describe a single line, and all points on that line would be solutions. This means there are infinitely many solutions to the system, not just \((2,3)\) and \((-3,5)\).

### Conclusion:
The system has infinitely many solutions because both equations describe the same line.

Would you like more details on how this works or have any other questions?

Here are 5 related questions to explore:
1. How do you determine if two lines are parallel or coincident from their equations?
2. What is the geometric interpretation of the solution to a system of two linear equations?
3. How can we find the equation of a line given two points?
4. What happens if two lines intersect at only one point? How do we solve such systems?
5. How can we solve a system of linear equations using substitution or elimination methods?

**Tip:** When solving systems of equations, always check if the equations might represent the same line by comparing their slopes and intercepts.

If (2,3) and (-3,5) are both solutions to a system of two linear equations, does the system have any other solutions? Explain.

Explore whether a system of two linear equations has any other solutions given two points. This problem delves into key algebraic concepts like systems of linear equations, the geometry of lines, and the slope formula. Suitable for Grades 9-11.

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