Let's analyze the problem:

• We are asked to find the equation of a line that is parallel to line $n$ and passes through point $P$.
• Two lines are parallel if they have the same slope.

Step 1: Determine the slope of line $n$

We can determine the slope by using the two points where line $n$ crosses the grid:

1. Line $n$ passes through the point $(6, -8)$.
2. Line $n$ also passes through the point $(0, 10)$.

The slope $m$ of a line is calculated using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Using the points $(6, -8)$ and $(0, 10)$: $m = \frac{-8 - 10}{6 - 0} = \frac{-18}{6} = -3$ So, the slope of line $n$ is $m = -3$.

Step 2: Equation of the line parallel to $n$

Since the line we want is parallel to line $n$, it will have the same slope of $m = -3$.

Step 3: Use point $P$ to find the equation

Point $P$ is located at $(-3, 1)$. We will use the point-slope form of the equation of a line: $y - y_1 = m(x - x_1)$ Substituting $m = -3$ and the coordinates of point $P$ $(-3, 1)$: $y - 1 = -3(x + 3)$ Simplifying: $y - 1 = -3x - 9$ $y = -3x - 8$

Step 4: Check the options

None of the options provided match exactly, but the closest match to this form is option D: $y = -3x - 6$. However, based on the calculation above, the correct equation should be $y = -3x - 8$.

Would you like to explore this discrepancy or need further clarification?

Relative Questions:

1. How do you calculate the slope of a line given two points?
2. What is the significance of the slope in parallel lines?
3. How do you convert a point-slope form equation into slope-intercept form?
4. What steps can be taken to verify if two lines are truly parallel?
5. How do small errors in slope or intercept affect the graph of a line?

Tip: Always recheck points used for slope calculations; even a small coordinate mistake can lead to incorrect conclusions.