The system of equations {3x – 5y = 2 and –6x + 10y = 7} has solutions.
Select . . . ▾
A. 0
B. 1
C. infinite

To determine the number of solutions for the system of equations:

\[
\begin{aligned}
1.\ & 3x - 5y = 2 \quad \text{(Equation 1)} \\
2.\ & -6x + 10y = 7 \quad \text{(Equation 2)}
\end{aligned}
\]

We can analyze the system by examining the relationships between the equations. Let's first simplify Equation 2 by dividing it by -2:

\[
-6x + 10y = 7 \implies 3x - 5y = -\frac{7}{2} \quad \text{(Simplified Equation 2)}
\]

Now, compare the simplified version of Equation 2 with Equation 1:

- **Equation 1:** \(3x - 5y = 2\)
- **Simplified Equation 2:** \(3x - 5y = -\frac{7}{2}\)

These two equations have the same left-hand side (\(3x - 5y\)) but different right-hand sides (2 and \(-\frac{7}{2}\)). This implies that the system of equations represents two parallel lines, which do not intersect.

Since parallel lines never intersect, there is no solution to this system of equations.

### Answer: **A. 0**

Would you like to explore why parallel lines do not intersect? Here are some related questions to think about:

1. How can you determine if two lines are parallel just by looking at their equations?
2. What happens when two lines in a system of equations have the same slope but different y-intercepts?
3. How does the elimination method help identify the number of solutions in a system of equations?
4. Can you give an example of a system of equations with infinite solutions?
5. How do systems of equations relate to real-world problems, such as finding the intersection point of two roads?

**Tip:** When dealing with systems of equations, always compare the slopes and y-intercepts first to understand the relationship between the lines!

Explore the system of equations 3x – 5y = 2 and –6x + 10y = 7 and understand why they have no solutions. This problem involves analyzing parallel lines in systems of linear equations, a fundamental concept in high school algebra.

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