Which line passes through the point (4, -7) and is perpendicular to the line x- 4y = 8? A. y = x-11 4 C. ソ=ー-x-6 B. y = -4x-23 D. y = -4x +9

Algebra and Trigonometry (6th Edition)
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Author:Robert F. Blitzer
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Question:**
Which line passes through the point (4, -7) and is perpendicular to the line \( x - 4y = 8 \)?

**Options:**
A. \( y = -\frac{1}{4}x - 11 \)

B. \( y = -4x - 23 \)

C. \( y = -\frac{1}{4}x - 6 \)

D. \( y = -4x + 9 \)

**Explanation:**
To find the line that passes through the given point and is perpendicular to the given line, we need to follow these steps:

1. **Convert the given line to slope-intercept form (y = mx + b):**
   The given line equation is \( x - 4y = 8 \).
   Solving for \( y \):
   \[
   x - 8 = 4y \\
   y = \frac{1}{4}x - 2
   \]
   Hence, the slope of the line \( x - 4y = 8 \) is \( \frac{1}{4} \).

2. **Determine the slope of the perpendicular line:**
   The slope of a line perpendicular to a line with slope \( m \) is \( -\frac{1}{m} \).
   Therefore, the slope of the line perpendicular to \( \frac{1}{4} \) is \( -4 \).

3. **Use the point (4, -7) and the perpendicular slope (-4) to find the line equation:**
   Using the point-slope form \( y - y_1 = m(x - x_1) \):
   \[
   y + 7 = -4(x - 4) \\
   y + 7 = -4x + 16 \\
   y = -4x + 9
   \]

   So, the equation of the line passing through the point (4, -7) and perpendicular to \( x - 4y = 8 \) is \( y = -4x + 9 \).

Therefore, the correct answer is:

**D. \( y = -4x + 9 \)**
Transcribed Image Text:**Question:** Which line passes through the point (4, -7) and is perpendicular to the line \( x - 4y = 8 \)? **Options:** A. \( y = -\frac{1}{4}x - 11 \) B. \( y = -4x - 23 \) C. \( y = -\frac{1}{4}x - 6 \) D. \( y = -4x + 9 \) **Explanation:** To find the line that passes through the given point and is perpendicular to the given line, we need to follow these steps: 1. **Convert the given line to slope-intercept form (y = mx + b):** The given line equation is \( x - 4y = 8 \). Solving for \( y \): \[ x - 8 = 4y \\ y = \frac{1}{4}x - 2 \] Hence, the slope of the line \( x - 4y = 8 \) is \( \frac{1}{4} \). 2. **Determine the slope of the perpendicular line:** The slope of a line perpendicular to a line with slope \( m \) is \( -\frac{1}{m} \). Therefore, the slope of the line perpendicular to \( \frac{1}{4} \) is \( -4 \). 3. **Use the point (4, -7) and the perpendicular slope (-4) to find the line equation:** Using the point-slope form \( y - y_1 = m(x - x_1) \): \[ y + 7 = -4(x - 4) \\ y + 7 = -4x + 16 \\ y = -4x + 9 \] So, the equation of the line passing through the point (4, -7) and perpendicular to \( x - 4y = 8 \) is \( y = -4x + 9 \). Therefore, the correct answer is: **D. \( y = -4x + 9 \)**
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