https://file.aiquickdraw.com/mathbot/file/1730936656287y26te69e.jpg

To solve this problem, let's go through the steps required to find the equation of a line perpendicular to the given line and passing through a specified point.

The given line equation is:
\[
y = 3x - 1
\]

### Step 1: Find the Slope of the Given Line
For the line \( y = 3x - 1 \), the slope (\( m \)) is the coefficient of \( x \), which is 3.

### Step 2: Determine the Slope of the Perpendicular Line
The slope of a line perpendicular to another is the negative reciprocal of the original line’s slope. Therefore, if the slope of the given line is 3, the slope of the perpendicular line will be:
\[
m_{\text{perpendicular}} = -\frac{1}{3}
\]

### Step 3: Use the Point-Slope Form
Now, we want the equation of a line with slope \( -\frac{1}{3} \) that passes through the point \( (3, 3) \). The point-slope form of a line is:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. Substituting \( m = -\frac{1}{3} \), \( x_1 = 3 \), and \( y_1 = 3 \), we get:
\[
y - 3 = -\frac{1}{3}(x - 3)
\]

### Step 4: Simplify the Equation
Distribute the slope on the right side:
\[
y - 3 = -\frac{1}{3}x + 1
\]
Now, add 3 to both sides to isolate \( y \):
\[
y = -\frac{1}{3}x + 4
\]

### Final Answer
The equation of the line perpendicular to \( y = 3x - 1 \) and passing through the point \( (3, 3) \) is:
\[
y = -\frac{1}{3}x + 4
\]

Would you like additional details on any part of this solution, or do you have further questions?

Here are some related questions for further exploration:
1. How do we derive the point-slope form of a line?
2. Why is the slope of a perpendicular line the negative reciprocal?
3. How would the solution change if the point was different, for example, \( (5, 2) \)?
4. Can we write this final answer in standard form instead of slope-intercept form?
5. What are other methods for finding the equation of a perpendicular line?

**Tip:** Always double-check the signs when calculating perpendicular slopes, as flipping or inverting incorrectly can lead to errors.

Write an equation for a line perpendicular to y = 3x - 1 and passing through the point (3,3).

Find the equation of a line perpendicular to y = 3x - 1 and passing through the point (3,3). This problem covers concepts of linear equations, slopes of perpendicular lines, and uses point-slope form. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation