https://file.aiquickdraw.com/mathbot/file/1731024819795x3yyxyv3.jpg

To solve this problem, let's break it down step-by-step.

1. **Identify the slope of line \( k \):**
   The equation of line \( k \) is given in point-slope form:
   \[
   y - 10 = -\frac{2}{5}(x - 10)
   \]
   In this form, the slope \( m \) is \(-\frac{2}{5}\).

2. **Find the slope of line \( \ell \):**
   Since line \( \ell \) is perpendicular to line \( k \), the slope of \( \ell \) will be the negative reciprocal of the slope of \( k \).
   - The slope of \( k \) is \(-\frac{2}{5}\), so the negative reciprocal is \(\frac{5}{2}\).
   - Therefore, the slope of \( \ell \) is \(\frac{5}{2}\).

3. **Use the point-slope form for line \( \ell \):**
   Line \( \ell \) passes through the point \( (4, 3) \) and has a slope of \(\frac{5}{2}\).
   - Using the point-slope form of a line, \( y - y_1 = m(x - x_1) \), we get:
     \[
     y - 3 = \frac{5}{2}(x - 4)
     \]

4. **Convert to slope-intercept form \( y = mx + b \):**
   - Expand the equation:
     \[
     y - 3 = \frac{5}{2}x - \frac{5}{2} \cdot 4
     \]
     \[
     y - 3 = \frac{5}{2}x - 10
     \]
   - Add 3 to both sides to isolate \( y \):
     \[
     y = \frac{5}{2}x - 7
     \]

5. **Final Answer:**
   The equation of line \( \ell \) in slope-intercept form is:
   \[
   y = \frac{5}{2}x - 7
   \]

Would you like a further breakdown of any part of this solution?

Here are some related questions you might consider:
1. What is the point-slope form of a line, and how does it differ from slope-intercept form?
2. How can you find the equation of a line parallel to a given line through a specific point?
3. Why is the slope of a perpendicular line the negative reciprocal?
4. What are other forms of a line's equation, such as the standard form?
5. How do you convert between different forms of a line equation?

**Tip:** Remember that perpendicular lines always have slopes that are negative reciprocals of each other.

Line k has an equation of y - 10 = -2/5(x - 10). Line l includes the point (4, 3) and is perpendicular to line k. What is the equation of line l?

Find the equation of a line perpendicular to y - 10 = -2/5(x - 10) that passes through the point (4, 3). This problem explores slope-intercept form, point-slope form, and perpendicular slopes. Ideal for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation