The problem asks to write the equation of a line that is parallel to $y = -4x + 7$ and passes through the point $C(4, 5)$.

Steps to solve:

1. Identify the slope of the given line: The slope of the line $y = -4x + 7$ is $-4$ (since the equation is in slope-intercept form $y = mx + b$, where $m$ is the slope).

2. Use the point-slope form: To find the equation of a line parallel to this one, we need to use the same slope ($-4$) and the point $C(4, 5)$. The point-slope form of a line is: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is the point and $m$ is the slope.

   Substituting $m = -4$, $x_1 = 4$, and $y_1 = 5$: $y - 5 = -4(x - 4)$

3. Simplify the equation: Expanding the right-hand side: $y - 5 = -4x + 16$ Add 5 to both sides: $y = -4x + 21$

Final Answer:

The equation of the line parallel to $y = -4x + 7$ and passing through the point $C(4, 5)$ is: $y = -4x + 21$

Would you like further details on any of these steps or have additional questions?

Here are some related questions:

1. What is the general form of the equation of a line?
2. How do you find the equation of a line perpendicular to a given line?
3. What is the significance of the slope in the equation of a line?
4. How would the equation change if the line passed through a different point?
5. How can we verify that two lines are parallel?

Tip: To quickly identify the slope of a line from an equation in slope-intercept form, simply look at the coefficient of $x$.