To find the equation of a line passing through the point $(1, -9)$ that is parallel to the line $y = -3x + 7$, follow these steps:

Step 1: Identify the slope of the given line

The given line is $y = -3x + 7$. This equation is in the slope-intercept form $y = mx + c$, where $m$ is the slope.

Here, the slope $m$ of the given line is $-3$.

Step 2: Use the point-slope form of the equation

The equation of a line parallel to the given line will have the same slope $-3$. The point-slope form of the equation of a line is:

$y - y_1 = m(x - x_1)$

where $(x_1, y_1)$ is the point the line passes through, and $m$ is the slope.

Substitute $m = -3$, $x_1 = 1$, and $y_1 = -9$:

$y - (-9) = -3(x - 1)$

Simplify the equation:

$y + 9 = -3x + 3$

Step 3: Write the equation in slope-intercept form

Subtract 9 from both sides to solve for $y$:

$y = -3x + 3 - 9$ $y = -3x - 6$

So, the equation of the line passing through the point $(1, -9)$ that is parallel to the line $y = -3x + 7$ is:

$\boxed{y = -3x - 6}$

Would you like more details or have any questions? Here are five related questions:

1. How do you find the equation of a line perpendicular to $y = -3x + 7$ passing through $(1, -9)$?
2. What is the distance between the line $y = -3x - 6$ and the origin?
3. How do you find the point of intersection between the lines $y = -3x - 6$ and $y = -3x + 7$?
4. What would be the equation of a line with the same slope but passing through the origin?
5. How does the y-intercept of a line affect its position on the graph?

Tip: When dealing with parallel lines, remember that their slopes are always equal. This is a key concept in identifying or constructing parallel lines in coordinate geometry.