https://file.aiquickdraw.com/mathbot/file/17369635318731osucwxt.jpg

Let’s solve the two parts of the question.

---

### **Part A:**
The equation of line \( j \) is given as \( y = 2x + 8 \).  
Another line \( k \) is **perpendicular** to \( j \) and passes through the point \( (6, -6) \).  

**Step 1:** Determine the slope of line \( k \).  
The slope of line \( j \) is \( 2 \). For perpendicular lines, the slopes are negative reciprocals.  
Thus, the slope of line \( k \) is:  
\[
m_k = -\frac{1}{2}
\]

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

Thus, the equation of line \( k \) is:
\[
y = -\frac{1}{2}x - 3
\]

The correct answer is **C**.

---

### **Part B:**
We are told that line \( m \) has a slope of \( -3 \). Another line \( n \) is **parallel** to \( m \).

**Key Concept:** Parallel lines have the same slope.  
Thus, the slope of line \( n \) is:
\[
\text{slope of } n = -3
\]

---

### Final Answers:
1. **Part A:** The equation of line \( k \) is \( y = -\frac{1}{2}x - 3 \) (Answer C).  
2. **Part B:** The slope of line \( n \) is \( -3 \).

---

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.

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?

Learn how to determine the equation of a line perpendicular to another line and passing through a given point. Understand the relationship between slopes of perpendicular and parallel lines. Includes a step-by-step solution suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation