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

The equation of a line can be written in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. 

To find the equation of a line parallel to line \( n \), we first need to determine the slope of line \( n \). Parallel lines have the same slope, so we will use the slope of line \( n \) for our new line.

### Step 1: Find the slope of line \( n \)
To find the slope, we need to identify two points on the line \( n \). Based on the image, we can see that line \( n \) passes through:
- Point \( (0, 6) \)
- Point \( (6, -4) \)

The slope formula is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substitute the coordinates of the points:
\[
m = \frac{-4 - 6}{6 - 0} = \frac{-10}{6} = -\frac{5}{3}
\]
So, the slope of line \( n \) is \( -\frac{5}{3} \).

### Step 2: Use the point-slope form to find the equation of the parallel line
The parallel line passes through point \( P \), which is approximately \( (-2, 2) \) based on the graph.

We use the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -\frac{5}{3} \) and the coordinates of point \( P(-2, 2) \):
\[
y - 2 = -\frac{5}{3}(x + 2)
\]
Simplify the equation:
\[
y - 2 = -\frac{5}{3}x - \frac{10}{3}
\]
Add 2 to both sides:
\[
y = -\frac{5}{3}x - \frac{10}{3} + 2
\]
Convert 2 to a fraction with denominator 3:
\[
y = -\frac{5}{3}x - \frac{10}{3} + \frac{6}{3}
\]
Simplify:
\[
y = -\frac{5}{3}x - \frac{4}{3}
\]

Thus, the equation of the line parallel to \( n \) and passing through point \( P \) is:
\[
y = -\frac{5}{3}x - \frac{4}{3}
\]

Would you like any further details or clarifications on this solution?

Here are 5 questions related to this topic for further exploration:
1. How would the equation change if the line were perpendicular instead of parallel?
2. How can you find the equation of a line given its slope and a point it passes through?
3. What are the general properties of parallel lines in a plane?
4. How can we use the slope-intercept form to quickly identify key characteristics of a line?
5. Can you describe the relationship between slope and angle of inclination of a line?

**Tip:** Always remember that parallel lines have the same slope, which can simplify finding equations of such lines through different points!

Find the equation of a line parallel to line n that passes through point P. This problem involves concepts such as slope, point-slope form, and parallel lines, suitable for students in Grades 8-10. The detailed solution provides step-by-step instructions.

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