To solve this problem, we need to write the equation of a line parallel to $y = -2x + 3$ that passes through the point $C(1, 5)$. Here's the solution step-by-step:

1. Identify the slope of the given line:
   The equation $y = -2x + 3$ is in slope-intercept form ($y = mx + b$), where $m$ is the slope.

   • Slope of the given line: $m = -2$.

2. Understand the condition for parallel lines:
   Parallel lines have the same slope. Therefore, the slope of the required line is also $m = -2$.

3. Use the point-slope form of a line:
   The equation of a line passing through a point $(x_1, y_1)$ with slope $m$ is:
   $y - y_1 = m(x - x_1)$
   Substituting $m = -2$, $x_1 = 1$, and $y_1 = 5$:
   $y - 5 = -2(x - 1)$

4. Simplify to slope-intercept form ($y = mx + b$):
   Distribute $-2$:
   $y - 5 = -2x + 2$
   Add 5 to both sides:
   $y = -2x + 7$

Final Answer:

The equation of the required line is: $y = -2x + 7$

Would you like further clarification or additional examples? Here are related questions to expand your understanding:

1. How do you find the equation of a line perpendicular to a given line passing through a point?
2. What are the characteristics of parallel lines in a coordinate plane?
3. How does the slope-intercept form compare to the point-slope form of a line?
4. How can you determine if two lines are parallel given their equations?
5. What happens to the slope of a line if it is rotated by 90 degrees?

Tip: Always remember, lines that are parallel have equal slopes, while perpendicular lines have slopes that are negative reciprocals of each other.