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.