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The question asks to find the equation of a line that passes through the point \((5, 5)\) and is perpendicular to the line given by the equation:

\[
y = \frac{1}{4}x + 10
\]

### Step 1: Find the slope of the perpendicular line
The slope of the given line is \(\frac{1}{4}\). The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, the slope \(m\) of the line we are looking for is:

\[
m = -\frac{1}{\frac{1}{4}} = -4
\]

### Step 2: Use the point-slope form of the line equation
The point-slope form of a line's equation is:

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

where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. We know the line passes through the point \((5, 5)\) and has a slope of \(-4\). Substituting into the point-slope form:

\[
y - 5 = -4(x - 5)
\]

### Step 3: Simplify the equation
Now, simplify the equation:

\[
y - 5 = -4x + 20
\]

\[
y = -4x + 25
\]

### Conclusion:
The equation of the line is \(y = -4x + 25\), which corresponds to option **C**.

Would you like a deeper explanation or any clarifications?

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Here are 5 related questions to further practice:
1. What is the equation of a line perpendicular to \(y = 3x + 2\) passing through \((2, -3)\)?
2. How do you find the slope of a line given its equation?
3. What is the equation of a line parallel to \(y = \frac{1}{2}x - 7\) and passing through \((0, 4)\)?
4. How do you determine if two lines are perpendicular based on their slopes?
5. Can two lines be both parallel and perpendicular?

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

What is the equation of the line that passes through point (5, 5) and perpendicular to y = (1/4)x + 10?

Learn how to find the equation of a line that passes through (5, 5) and is perpendicular to y = (1/4)x + 10. This problem involves calculating the slope of the perpendicular line and applying the point-slope form equation. Suitable for students in Grades 9-11.

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