Question 8 of 10 ( 1 point) । Question Attempt: 1 of 1 Consider the line $$ y=9 x-8 $$. Find the equation of the line that is perpendicular to this line and passes through the point $$ (5,-5) $$. Find the equation of the line that is parallel to this line and passes through the point $$ (5,-5) $$. Note that the ALEKS graphing calculator may be helpful in checking your answer. Equation of parallel line:

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To find the equations of the lines that are perpendicular and parallel to the given line $$ y = 9x - 8 $$ and pass through the point $$ (5, -5) $$, we will follow these steps:

### Step 1: Identify the slope of the given line
The given line is in the slope-intercept form $$ y = mx + b $$, where $$ m $$ is the slope. From the equation $$ y = 9x - 8 $$, we can see that the slope $$ m $$ is $$ 9 $$.

### Step 2: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope $$ m_{\perp} $$ of the perpendicular line is:
$$
m_{\perp} = -\frac{1}{m} = -\frac{1}{9}
$$

### Step 3: Use the point-slope form to find the equation of the perpendicular line
The point-slope form of a line is given by:
$$
y - y_1 = m(x - x_1)
$$
where $$ (x_1, y_1) $$ is a point on the line. Here, we have the point $$ (5, -5) $$ and the slope $$ m_{\perp} = -\frac{1}{9} $$.

Substituting these values into the point-slope form:
$$
y - (-5) = -\frac{1}{9}(x - 5)
$$
This simplifies to:
$$
y + 5 = -\frac{1}{9}x + \frac{5}{9}
$$
Now, we can isolate $$ y $$:
$$
y = -\frac{1}{9}x + \frac{5}{9} - 5
$$
$$
y = -\frac{1}{9}x + \frac{5}{9} - \frac{45}{9}
$$
$$
y = -\frac{1}{9}x - \frac{40}{9}
$$

### Step 4: Find the equation of the parallel line
The slope of the parallel line is the same as the original line, which is $$ 9 $$. Using the point-slope form again with the point $$ (5, -5) $$:
$$
y - (-5) = 9(x - 5)
$$
This simplifies to:
$$
y + 5 = 9x - 45
$$
Now, isolating $$ y $$:
$$
y = 9x - 45 - 5
$$
$$
y = 9x - 50
$$

### Summary of Results
- The equation of the perpendicular line is:
$$
y = -\frac{1}{9}x - \frac{40}{9}
$$
- The equation of the parallel line is:
$$
y = 9x - 50
$$

Thus, the equation of the parallel line is $$ y = 9x - 50 $$.
The equation of the parallel line is $$ y = 9x - 50 $$.

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