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?

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!