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We need to find the equation of the line passing through the point \((-1, 4)\) and perpendicular to the line \(x - 4y = 3\).

### Step 1: Rewrite the given equation in slope-intercept form
Start by rewriting the equation of the line \(x - 4y = 3\) in the form \(y = mx + b\), where \(m\) is the slope:

\[
x - 4y = 3
\]
Subtract \(x\) from both sides:
\[
-4y = -x + 3
\]
Now, divide by \(-4\):
\[
y = \frac{1}{4}x - \frac{3}{4}
\]
Thus, the slope of this line is \(m = \frac{1}{4}\).

### Step 2: Find the slope of the perpendicular line
The slopes of two perpendicular lines are negative reciprocals of each other. So, if the slope of the original line is \(m = \frac{1}{4}\), the slope of the line perpendicular to it will be:

\[
m_{\text{perpendicular}} = -\frac{1}{\frac{1}{4}} = -4
\]

### Step 3: Use the point-slope form
Now, we use the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope. Here, the point is \((-1, 4)\) and the slope is \(-4\). Plug these values into the equation:

\[
y - 4 = -4(x + 1)
\]
Distribute the \(-4\) on the right:
\[
y - 4 = -4x - 4
\]
Add 4 to both sides:
\[
y = -4x
\]

### Final Answer
The equation of the line \(D\) passing through the point \((-1, 4)\) and perpendicular to the line \(x - 4y = 3\) is:
\[
y = -4x
\]

Would you like further details or have any other questions?

Here are some related questions:
1. How do you find the equation of a line parallel to another?
2. What happens to the equation if a line passes through the origin?
3. Can two lines with identical slopes ever intersect?
4. How do you find the intersection point of two lines?
5. How do we handle vertical or horizontal lines in these types of problems?

**Tip:** When lines are perpendicular, always remember to flip the slope and change its sign!

Provide the equation for the line (D) passing through the point (-1, 4) and perpendicular to the line x - 4y = 3.

Find the equation of a line passing through the point (-1, 4) and perpendicular to the line x - 4y = 3. This problem covers key coordinate geometry concepts, including slopes of perpendicular lines and the point-slope form equation. Suitable for Grades 9-11 students.

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