Questions: Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form. Passing through (2,-6) and perpendicular to the line whose equation is y=1/4 x+1 Write an equation for the line in point-slope form. y+6=-4(x-2) (Simplify your answer. Use integers or fractions for any numbers in the equation.) Write an equation for the line in slope-intercept form. (Simplify your answer. Use integers or fractions for any numbers in the equation.)

Solution Steps

To solve this problem, we need to find the equation of a line that passes through a given point and is perpendicular to another given line. The steps are as follows:

1. Identify the slope of the given line: The given line is in the form \( y = \frac{1}{4}x + 1 \), so its slope is \( \frac{1}{4} \).

2. Find the perpendicular slope: The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the perpendicular slope is \( -4 \).

3. Use the point-slope form: With the perpendicular slope and the given point \((2, -6)\), use the point-slope form of a line equation: \( y - y_1 = m(x - x_1) \).

4. Convert to slope-intercept form: Simplify the point-slope equation to get the slope-intercept form \( y = mx + b \).

The given line is \( y = \frac{1}{4}x + 1 \). The slope of this line is \( \frac{1}{4} \).

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the perpendicular slope is \( -4 \).

We have a point \((2, -6)\) and a slope \( m = -4 \). The point-slope form of the line is: \[ y - y_1 = m(x - x_1) \] Substituting the values, we get: \[ y + 6 = -4(x - 2) \]

Simplify the point-slope equation to get the slope-intercept form: \[ y + 6 = -4x + 8 \] \[ y = -4x + 8 - 6 \] \[ y = -4x + 2 \]

Final Answer