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To find which line has the same slope as the line passing through points \((-4, -1)\) and \((-1, 5)\), let's calculate the slope of that line and compare it with the options.

### Step 1: Calculate the Slope
The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-4, -1)\) and \((-1, 5)\):
- \( x_1 = -4 \), \( y_1 = -1 \)
- \( x_2 = -1 \), \( y_2 = 5 \)

\[
m = \frac{5 - (-1)}{-1 - (-4)} = \frac{5 + 1}{-1 + 4} = \frac{6}{3} = 2
\]

So, the slope of the line passing through \((-4, -1)\) and \((-1, 5)\) is \(2\).

### Step 2: Analyze the Options

1. **Option 1:** \( y = \frac{1}{2}x + 5 \)
   - The slope here is \( \frac{1}{2} \), which is **not** equal to 2.

2. **Option 2:** \( y = 2 \)
   - This is a horizontal line with a slope of \(0\), which is **not** equal to 2.

3. **Option 3:** \( 6x - 3y = 5 \)
   - Let's rearrange this to find the slope.
   - Rewrite in slope-intercept form (\(y = mx + b\)):
     \[
     -3y = -6x + 5
     \]
     \[
     y = 2x - \frac{5}{3}
     \]
   - The slope here is \(2\), which **matches** the slope we calculated.

4. **Option 4:** The line passing through \((1, 4)\) and \((1, 6)\)
   - This line has undefined (infinite) slope because it is a vertical line (same \(x\)-coordinates for both points). This is **not** equal to 2.

### Conclusion
The correct answer is:
**Option 3:** \( 6x - 3y = 5 \)

Would you like a more detailed explanation, or do you have any other questions?

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Here are 5 follow-up questions that might deepen understanding:
1. How do you determine if two lines are parallel?
2. How can you find the slope of a line if given only the equation in standard form?
3. Why does a vertical line have an undefined slope?
4. What are the steps to convert from standard form to slope-intercept form?
5. How does the slope-intercept form of a line help in graphing?

**Tip:** Remember, parallel lines always have the same slope!

Which line has the same slope as the line passing through (-4, -1) and (-1, 5)?

Learn how to find the slope of a line that passes through two points and compare it with the slopes of other lines. This problem involves calculating slope using coordinate geometry and understanding parallel lines. Suitable for Grades 8-10.

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