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What is the equation of the line parallel to line n that passes through point P?
A.y=3x+12

B.y=−3x+3

C.y=3x−6

D.y=−3x−6


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.

What is the equation of the line parallel to line n that passes through point P?

Learn how to find the equation of a line parallel to line n that passes through point P. This problem covers linear equations, slopes, and parallel line properties, suitable for students in Grades 8-10.

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