Given the line equation \(2x + 6y - 7 = 0\), we can find its slope, and then find the slope of a line perpendicular to it. The process involves the following steps:

### Step 1: Determine the slope of the given line
First, rewrite the given equation in the slope-intercept form \(y = mx + b\):

\[ 2x + 6y - 7 = 0 \]
\[ 6y = -2x + 7 \]
\[ y = -\frac{1}{3}x + \frac{7}{6} \]

From the equation \(y = -\frac{1}{3}x + \frac{7}{6}\), we can see that the slope (\(m\)) of the given line is \(-\frac{1}{3}\).

### Step 2: Determine the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope of the line perpendicular to the given line is:
\[ m_{\perpendicular} = -\left(-\frac{1}{3}\right)^{-1} = 3 \]

### Step 3: Determine the equation of the perpendicular line
We need the equation of a line with slope \(3\) that passes through the point \(C\left(\frac{5}{4}, \frac{3}{4}\right)\).

Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]

Substitute \(y_1 = \frac{3}{4}\), \(x_1 = \frac{5}{4}\), and \(m = 3\):
\[ y - \frac{3}{4} = 3\left(x - \frac{5}{4}\right) \]

### Step 4: Simplify the equation
\[ y - \frac{3}{4} = 3x - \frac{15}{4} \]
\[ y = 3x - \frac{15}{4} + \frac{3}{4} \]
\[ y = 3x - \frac{12}{4} \]
\[ y = 3x - 3 \]

Thus, the equation of the line perpendicular to \(2x + 6y - 7 = 0\) and passing through \(C\left(\frac{5}{4}, \frac{3}{4}\right)\) is:
\[ y = 3x - 3 \]

Would you like further details or have any questions?

Here are some questions you might consider next:
1. How do you convert a linear equation from general form to slope-intercept form?
2. What is the point-slope form of a linear equation?
3. How can you find the coordinates of a point dividing a segment in a given ratio?
4. What are the properties of perpendicular lines in coordinate geometry?
5. How do you verify if two lines are perpendicular?

**Tip:** When working with equations of lines, always ensure to simplify and convert them into a familiar form (such as slope-intercept form) for easier interpretation and manipulation.

Learn how to find the equation of a line perpendicular to 2x + 6y - 7 = 0 passing through a specific point. This problem involves coordinate geometry concepts, including equations of lines and the perpendicular lines theorem. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation