Math Problem Statement

The equation for line j can be written as y = 2x + 8. Another line k is perpendicular to line j and passes through the point (6, -6). Choose the equation for line k. Line m has a slope of -3. Another line n is parallel to line m. What is the slope of line n?

Solution

Let’s solve the two parts of the question.


Part A:

The equation of line jj is given as y=2x+8y = 2x + 8.
Another line kk is perpendicular to jj and passes through the point (6,6)(6, -6).

Step 1: Determine the slope of line kk.
The slope of line jj is 22. For perpendicular lines, the slopes are negative reciprocals.
Thus, the slope of line kk is:
mk=12m_k = -\frac{1}{2}

Step 2: Find the equation of line kk.
The slope-intercept form of a line is: y=mx+by = mx + b Here m=12m = -\frac{1}{2}, and the line passes through (6,6)(6, -6). Substitute x=6x = 6, y=6y = -6, and m=12m = -\frac{1}{2} into the equation to solve for bb: 6=12(6)+b-6 = -\frac{1}{2}(6) + b 6=3+b-6 = -3 + b b=3b = -3

Thus, the equation of line kk is: y=12x3y = -\frac{1}{2}x - 3

The correct answer is C.


Part B:

We are told that line mm has a slope of 3-3. Another line nn is parallel to mm.

Key Concept: Parallel lines have the same slope.
Thus, the slope of line nn is: slope of n=3\text{slope of } n = -3


Final Answers:

  1. Part A: The equation of line kk is y=12x3y = -\frac{1}{2}x - 3 (Answer C).
  2. Part B: The slope of line nn is 3-3.

Would you like me to go into further detail about any of the steps? Here are some related questions:

  1. How do you find the slope of a perpendicular line?
  2. What is the significance of the y-intercept in a line equation?
  3. How are parallel and perpendicular lines related in geometry?
  4. Could you explain more about the negative reciprocal rule for slopes?
  5. What happens if the line does not pass through a given point?

Tip: Always verify whether a line equation satisfies the given conditions by substituting the point into the equation.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Equations
Slopes of Perpendicular Lines
Slopes of Parallel Lines

Formulas

Slope-intercept form: y = mx + b
Perpendicular slope relation: m1 * m2 = -1

Theorems

Slope of perpendicular lines
Slope of parallel lines

Suitable Grade Level

Grades 8-10