We are tasked with finding the equation of a line that passes through the point $(-3, 4)$ and is parallel to the line given by the equation $x - y = 9$.

Step-by-step Solution:

1. Find the slope of the given line: The equation of the given line is $x - y = 9$. Let's rewrite it in slope-intercept form $y = mx + b$, where $m$ is the slope.

   $x - y = 9$ Solving for $y$, we get: $y = x - 9$

   From this, we see that the slope $m$ of the line is 1.

2. Find the slope of the parallel line: Since the lines are parallel, they have the same slope. Therefore, the slope of the line we are looking for is also $m = 1$.

3. Use the point-slope form: The point-slope form of a line 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.

   Substituting $m = 1$ and the point $(-3, 4)$, we get: $y - 4 = 1(x + 3)$ Simplifying this equation: $y - 4 = x + 3$ $y = x + 7$

So, the equation of the line that passes through $(-3, 4)$ and is parallel to $x - y = 9$ is: $y = x + 7$

Would you like more details on this solution or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you find the equation of a line perpendicular to another line?
2. What is the slope-intercept form of a line, and why is it useful?
3. Can you find the equation of a line that passes through two points?
4. How does the point-slope form of a line differ from the slope-intercept form?
5. What is the general form of a linear equation, and how do you convert between forms?

Tip: When finding equations of parallel lines, remember that they always have identical slopes!