Write the point-slope form of the line that passes through \( (5,5) \) and is perpendlcular to a line with a slope of \( \frac{1}{4} \). Include all of your work in your final answer. Type your answer in the box provided to submit your solution.

To write the point-slope form of the line that passes through the point \( (5,5) \) and is perpendicular to a line with a slope of \( \frac{1}{4} \), we need to follow these steps: 1. Determine the slope of the line that is perpendicular to the given line. 2. Use the point-slope form of a line to write the equation. Step 1: Determine the slope of the perpendicular line The slope of the given line is \( \frac{1}{4} \). The slope of a line perpendicular to this one is the negative reciprocal of \( \frac{1}{4} \). The negative reciprocal is found by flipping the fraction and changing the sign, so the slope of the perpendicular line is \( -4 \). Step 2: Use the point-slope form to write the equation The point-slope form of a line is given by the equation: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope of the line and \( (x_1, y_1) \) is a point on the line. We have the point \( (5,5) \) and the slope \( m = -4 \). Plugging these values into the point-slope form, we get: \[ y - 5 = -4(x - 5) \] Now, let's simplify this equation to get it into the point-slope form: \[ y - 5 = -4x + 20 \] So, the point-slope form of the line that passes through \( (5,5) \) and is perpendicular to a line with a slope of \( \frac{1}{4} \) is: \[ y = -4x + 25 \]