Consider the line $$ y=6 x-7 $$. Find the equation of the line that is parallel to this line and passes through the point $$ (8,-3) $$. Find the equation of the line that is perpendicular to this line and passes through the point $$ (8,-3) $$. Note that the ALEKS graphing calculator may be helpful in checking your answer. Equation of parallel line: Equation of perpendicular line: $$ y=\frac{1}{\square} $$

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To solve the problem, we need to find two lines based on the given line $$ y = 6x - 7 $$ and the point $$ (8, -3) $$.

### Step 1: Find the equation of the parallel line

1. **Identify the slope of the given line**: The slope $$ m $$ of the line $$ y = 6x - 7 $$ is $$ 6 $$.
2. **Use the point-slope form of the line equation**: The point-slope form is given by:
   $$
   y - y_1 = m(x - x_1)
   $$
   where $$ (x_1, y_1) $$ is the point through which the line passes, and $$ m $$ is the slope.
3. **Substituting the values**: Here, $$ m = 6 $$, $$ x_1 = 8 $$, and $$ y_1 = -3 $$:
   $$
   y - (-3) = 6(x - 8)
   $$
   Simplifying this:
   $$
   y + 3 = 6x - 48
   $$
   $$
   y = 6x - 51
   $$

### Step 2: Find the equation of the perpendicular line

1. **Identify the slope of the perpendicular line**: The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line is:
   $$
   m_{\perpendicular} = -\frac{1}{6}
   $$
2. **Use the point-slope form again**: Using the same point $$ (8, -3) $$:
   $$
   y - (-3) = -\frac{1}{6}(x - 8)
   $$
   Simplifying this:
   $$
   y + 3 = -\frac{1}{6}x + \frac{8}{6}
   $$
   $$
   y + 3 = -\frac{1}{6}x + \frac{4}{3}
   $$
   $$
   y = -\frac{1}{6}x + \frac{4}{3} - 3
   $$
   $$
   y = -\frac{1}{6}x + \frac{4}{3} - \frac{9}{3}
   $$
   $$
   y = -\frac{1}{6}x - \frac{5}{3}
   $$

### Final Answers

- **Equation of the parallel line**: $$ y = 6x - 51 $$
- **Equation of the perpendicular line**: $$ y = -\frac{1}{6}x - \frac{5}{3} $$

Thus, the answer for the perpendicular line in the required format is:
$$
y = \frac{1}{-6}
$$

So, the final answers are:
- Equation of parallel line: $$ y = 6x - 51 $$
- Equation of perpendicular line: $$ y = \frac{1}{-6} $$
- Equation of parallel line: $$ y = 6x - 51 $$ - Equation of perpendicular line: $$ y = -\frac{1}{6}x - \frac{5}{3} $$

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