The problem asks to find the equation of a line that passes through the point $(4, 6)$ and is parallel to the line given by the equation $x + 4y = 3$. The final equation should be in the slope-intercept form $y = mx + b$.

Step-by-Step Solution:

1. Find the slope of the given line:
   The given line is $x + 4y = 3$. To find the slope, we need to rewrite it in slope-intercept form $y = mx + b$, where $m$ is the slope.

   Start by solving for $y$: $x + 4y = 3 \quad \Rightarrow \quad 4y = -x + 3 \quad \Rightarrow \quad y = -\frac{1}{4}x + \frac{3}{4}$ So, the slope of the given line is $m = -\frac{1}{4}$.

2. Use the same slope for the parallel line:
   Since parallel lines have the same slope, the equation of the line passing through $(4, 6)$ will also have a slope of $-\frac{1}{4}$. Therefore, the equation of the line is of the form: $y = -\frac{1}{4}x + b$ where $b$ is the y-intercept that we need to find.

3. Substitute the point $(4, 6)$ into the equation:
   To find $b$, substitute $x = 4$ and $y = 6$ into the equation: $6 = -\frac{1}{4}(4) + b \quad \Rightarrow \quad 6 = -1 + b \quad \Rightarrow \quad b = 7$

4. Write the final equation:
   Now that we know $b = 7$, the equation of the line is: $y = -\frac{1}{4}x + 7$

Thus, the equation of the line that passes through $(4, 6)$ and is parallel to $x + 4y = 3$ is: $y = -\frac{1}{4}x + 7$

Would you like more details on any step? Here are some related questions:

1. How do you determine the slope of a line given in standard form?
2. Why do parallel lines have the same slope?
3. How do you convert an equation to slope-intercept form?
4. What is the difference between parallel and perpendicular lines in terms of slopes?
5. How can you find the equation of a line given two points?

Tip: Always express the equation of a line in slope-intercept form when solving for slope-related problems, as it makes it easier to compare and find parallel or perpendicular lines.