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.)

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

Transcript text: 2.4 -More on Slope Question 2, 2.4.7 Part 2 of 2 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 $y=\frac{1}{4} x+1$ 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

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:

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} $.
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 $.
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) $.
Convert to slope-intercept form: Simplify the point-slope equation to get the slope-intercept form $ y = mx + b $.

Step 1: Identify the Slope of the Given Line

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

Step 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 $.

Step 3: Use the Point-Slope Form

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: \[ y + 6 = -4(x - 2) \]

Step 4: Convert to Slope-Intercept Form

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