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The problem asks to find the equation of a line that passes through the point \((-1, 7)\) and is perpendicular to the line given by the equation \(4x + y = 11\).

### Step-by-Step Solution:

1. **Find the slope of the given line \(4x + y = 11\):**

   Rewrite the equation in slope-intercept form, \(y = mx + b\), where \(m\) is the slope.

   \[
   y = -4x + 11
   \]

   The slope of this line (\(m_1\)) is \(-4\).

2. **Determine the slope of the perpendicular line:**

   If two lines are perpendicular, their slopes are negative reciprocals of each other. Thus, the slope of the perpendicular line (\(m_2\)) is:

   \[
   m_2 = \frac{1}{4}
   \]

3. **Formulate the equation of the perpendicular line passing through the point \((-1, 7)\):**

   Using the point-slope form of a line equation:

   \[
   y - y_1 = m(x - x_1)
   \]

   Substituting \(m = \frac{1}{4}\) and the point \((-1, 7)\):

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

4. **Simplify the equation:**

   Distribute and simplify:

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

   Add 7 to both sides:

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

   \[
   y = \frac{1}{4}x + \frac{1}{4} + \frac{28}{4}
   \]

   \[
   y = \frac{1}{4}x + \frac{29}{4}
   \]

5. **Final Answer:**

   The equation of the line that passes through the point \((-1, 7)\) and is perpendicular to the line \(4x + y = 11\) is:

   \[
   y = \frac{1}{4}x + \frac{29}{4}
   \]

Would you like more details or have any questions?

### Further Related Questions:

1. How do you determine if two lines are parallel or perpendicular?
2. What is the general form of a line equation, and how can it be derived from point-slope form?
3. Can you provide examples of finding the equation of a line given different points and slopes?
4. What are some methods to find the intersection point of two lines?
5. How do you convert between different forms of linear equations (point-slope, slope-intercept, standard)?

### Tip:

When dealing with perpendicular lines, remember that their slopes are negative reciprocals. This is a quick way to find the slope of a line perpendicular to a given line.

Find the equation of the line which passes through the point (-1, 7) and is perpendicular to the line with equation 4x + y = 11.

Learn how to find the equation of a line that passes through the point (-1, 7) and is perpendicular to the line 4x + y = 11. This problem involves key algebraic concepts such as linear equations, slopes, and the perpendicular line theorem. Suitable for students in Grades 9-10.

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