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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:
\[
y = -\frac{1}{4}x + 7
\]

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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.

Write an equation of the line containing the point (4,6) and parallel to the line x + 4y = 3.

Learn how to find the equation of a line that passes through the point (4,6) and is parallel to the line x + 4y = 3. This problem involves converting the standard form to slope-intercept form and applying key algebraic concepts. Suitable for Grades 8-10 students.

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