https://r2.erweima.ai/mathbot/image/17259581520566hplb53j.jpg

The problem asks to find the equation of a line perpendicular to the line \(4x - 3y + 6 = 0\), passing through the point \(M(2; 1)\).

### Step-by-step solution:

1. **Find the slope of the given line**:  
   The general form of a line equation is \(Ax + By + C = 0\), and its slope is given by \(\frac{-A}{B}\).  
   For the line \(4x - 3y + 6 = 0\), the slope \(m_1\) is:
   \[
   m_1 = \frac{-4}{-3} = \frac{4}{3}
   \]

2. **Find the slope of the perpendicular line**:  
   The slope of a line perpendicular to another is the negative reciprocal of the original slope.  
   Therefore, the slope \(m_2\) of the perpendicular line is:
   \[
   m_2 = -\frac{1}{\frac{4}{3}} = -\frac{3}{4}
   \]

3. **Write the equation of the perpendicular line**:  
   The equation of a line with slope \(m_2\) passing through point \(M(2; 1)\) can be written in point-slope form:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \((x_1, y_1) = (2, 1)\) and \(m = -\frac{3}{4}\):
   \[
   y - 1 = -\frac{3}{4}(x - 2)
   \]

4. **Simplify the equation**:  
   Expand and simplify:
   \[
   y - 1 = -\frac{3}{4}x + \frac{3}{2}
   \]
   \[
   y = -\frac{3}{4}x + \frac{3}{2} + 1
   \]
   \[
   y = -\frac{3}{4}x + \frac{5}{2}
   \]

So, the equation of the line perpendicular to \(4x - 3y + 6 = 0\) and passing through the point \(M(2; 1)\) is:
\[
y = -\frac{3}{4}x + \frac{5}{2}
\]

Would you like a detailed explanation on any specific step?

### Related questions:
1. How do you find the slope of a line from its equation?
2. What is the point-slope form of a line, and how is it used?
3. Why is the negative reciprocal used to find the slope of a perpendicular line?
4. How can you convert a line from point-slope form to slope-intercept form?
5. What happens when the slopes of two lines are neither perpendicular nor parallel?

**Tip**: The slope-intercept form \(y = mx + b\) is useful for quickly identifying the slope and intercept of a line.

Преподаватель попросил студента написать уравнение перпендикуляра к прямой 4x - 3y + 6 = 0, проходящего через точку M(2; 1). Что должен ответить студент?

Learn how to find the equation of a line perpendicular to 4x - 3y + 6 = 0, passing through point M(2, 1). This problem covers important concepts like finding slopes of perpendicular lines, using point-slope form, and simplifying linear equations. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation