https://file.aiquickdraw.com/mathbot/file/1727305248801b9oykpbn.jpg

The problem in the image asks to:

1. **Write an equation for a line passing through the point \((3,6)\) that is parallel to** \( y = \frac{3}{7}x + 1 \).
2. **Write a second equation for a line passing through the given point that is perpendicular to the given line.**

### Step 1: Equation for the Parallel Line
When two lines are parallel, they have the same slope. The given line is \( y = \frac{3}{7}x + 1 \), which has a slope \( m = \frac{3}{7} \).

To find the equation of the line parallel to this one, passing through the point \((3,6)\), we use the **point-slope form** of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) = (3,6) \) and \( m = \frac{3}{7} \).

Substitute these values into the point-slope form:
\[
y - 6 = \frac{3}{7}(x - 3)
\]
Simplify this:
\[
y - 6 = \frac{3}{7}x - \frac{9}{7}
\]
\[
y = \frac{3}{7}x - \frac{9}{7} + \frac{42}{7}
\]
\[
y = \frac{3}{7}x + \frac{33}{7}
\]
So, the equation of the line parallel to \( y = \frac{3}{7}x + 1 \) passing through \((3,6)\) is:
\[
y = \frac{3}{7}x + \frac{33}{7}
\]

### Step 2: Equation for the Perpendicular Line
For two lines to be perpendicular, the product of their slopes must be \(-1\). The slope of the given line is \( \frac{3}{7} \), so the slope of a line perpendicular to it is the negative reciprocal:
\[
m_{\perpendicular} = -\frac{7}{3}
\]

Using the point-slope form again, with \( m = -\frac{7}{3} \) and the point \((3,6)\):
\[
y - 6 = -\frac{7}{3}(x - 3)
\]
Simplify this:
\[
y - 6 = -\frac{7}{3}x + 7
\]
\[
y = -\frac{7}{3}x + 7 + 6
\]
\[
y = -\frac{7}{3}x + 13
\]
So, the equation of the line perpendicular to \( y = \frac{3}{7}x + 1 \) and passing through \((3,6)\) is:
\[
y = -\frac{7}{3}x + 13
\]

### Final Answer:
1. The equation of the line parallel to \( y = \frac{3}{7}x + 1 \) through \((3,6)\) is:
\[
y = \frac{3}{7}x + \frac{33}{7}
\]
2. The equation of the line perpendicular to \( y = \frac{3}{7}x + 1 \) through \((3,6)\) is:
\[
y = -\frac{7}{3}x + 13
\]

Let me know if you want more details or have any questions!

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Here are 5 related questions to explore:

1. How do you find the slope of a line when given two points?
2. What are the conditions for two lines to be parallel?
3. How do you derive the point-slope form of a line?
4. What happens to the slope when a line is perpendicular to another?
5. Can two perpendicular lines ever be parallel? Why or why not?

**Tip:** Always check the slopes when writing equations for parallel or perpendicular lines to ensure accuracy.

Write an equation for a line passing through the point (3,6) that is parallel to y = 3/7x + 1. Then write a second equation for a line passing through the given point that is perpendicular to the given line.

Learn how to find the equations of a line passing through (3,6) that is parallel to y = 3/7x + 1, and another line passing through the same point that is perpendicular to it. This problem explores linear equations and slopes, perfect for Grades 9-11.

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