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To find the exact distance between two parallel lines given by their equations, we can use the formula:

\[
\text{Distance} = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}
\]

where the lines are in the form \(Ax + By = C\). Here are the two equations:

1. \(3x - 5y = 2\)
2. \(6x - 10y = -7\)

### Step 1: Confirm Parallelism
First, let's ensure the lines are parallel by checking if their coefficients are proportional.

For the first line, \(A_1 = 3\), \(B_1 = -5\), and for the second line, \(A_2 = 6\), \(B_2 = -10\).

Since \(\frac{A_2}{A_1} = \frac{6}{3} = 2\) and \(\frac{B_2}{B_1} = \frac{-10}{-5} = 2\), the lines are parallel.

### Step 2: Apply the Distance Formula
Now we identify \(C_1\) and \(C_2\) from each equation:
- For \(3x - 5y = 2\), \(C_1 = 2\).
- For \(6x - 10y = -7\), \(C_2 = -7\).

The distance between the lines is:
\[
\text{Distance} = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} = \frac{|-7 - 2|}{\sqrt{3^2 + (-5)^2}}
\]

### Step 3: Calculate the Distance
1. Calculate \(|C_2 - C_1|\):
   \[
   |-7 - 2| = |-9| = 9
   \]

2. Calculate \(\sqrt{A^2 + B^2}\):
   \[
   \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}
   \]

3. Substitute and solve:
   \[
   \text{Distance} = \frac{9}{\sqrt{34}}
   \]

Thus, the exact distance between the two lines is:
\[
\frac{9}{\sqrt{34}}
\]

If you have further questions or need details on specific steps, feel free to ask!

### Related Questions
1. How do you derive the distance formula between two parallel lines?
2. What are the steps to convert equations into slope-intercept form for better visualization?
3. Can you explain why only parallel lines have a constant distance between them?
4. How would the distance calculation change if the lines were perpendicular?
5. What real-life applications require calculating the distance between parallel lines?

### Tip
Always check if the lines are parallel before calculating the distance. Non-parallel lines will not have a consistent distance between them.

Find the exact distance between the two lines: 3x - 5y = 2 and 10y = 6x - 7.

Learn how to find the exact distance between two parallel lines with equations 3x - 5y = 2 and 6x - 10y = -7. This solution explains the distance formula, verification of parallelism, and step-by-step calculations. Ideal for students in Grades 10-12.

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